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Solving Linear Inequalities

30 questions

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Question 1Question →

What is the complete set of real numbers xx that satisfy the inequality 2−x3−3x−14≥2\frac{2 - x}{3} - \frac{3x - 1}{4} \geq 2?

Show answer & explanation

Answer: x≤−1x \leq -1

Answer

The complete set of real numbers satisfying the inequality is x≤−1x \leq -1.
To solve the inequality 2−x3−3x−14≥2\frac{2 - x}{3} - \frac{3x - 1}{4} \geq 2, we first eliminate the denominators by multiplying both sides by their least common multiple, which is 1212. This yields 4(2−x)−3(3x−1)≥244(2 - x) - 3(3x - 1) \geq 24. Distributing the terms gives 8−4x−9x+3≥248 - 4x - 9x + 3 \geq 24. Combining like terms results in 11−13x≥2411 - 13x \geq 24. Subtracting 1111 from both sides yields −13x≥13-13x \geq 13. Finally, dividing by −13-13 requires reversing the inequality sign, leading to the solution x≤−1x \leq -1.

Step-by-Step Solution

1
Multiply all terms on both sides of the inequality by 1212, which is the least common multiple of the denominators 33 and 44.
4(2−x)−3(3x−1)≥244(2 - x) - 3(3x - 1) \geq 24
Multiplying by a positive number preserves the inequality direction while eliminating the fractions to simplify the equation.
2
Distribute the coefficients 44 and −3-3 to the terms inside the parentheses.
8−4x−9x+3≥248 - 4x - 9x + 3 \geq 24
Applying the distributive property expands the expression. Note that multiplying −3-3 by −1-1 results in +3+3.
3
Combine like terms on the left side of the inequality.
11−13x≥2411 - 13x \geq 24
Grouping the constants (8+3=118 + 3 = 11) and the variable terms (−4x−9x=−13x-4x - 9x = -13x) simplifies the inequality.
4
Subtract 1111 from both sides of the inequality.
−13x≥13-13x \geq 13
Isolating the variable term on the left side by moving the constant term to the right side.
5
Divide both sides of the inequality by −13-13 and reverse the direction of the inequality sign.
x≤−1x \leq -1
Dividing by a negative number requires reversing the inequality sign from ≥\geq to ≤\leq.

Key Concept

Solving linear inequalities by clearing fractions, distributing terms correctly, and reversing the inequality sign when multiplying or dividing by a negative number.
Estimated Time:2m 0s
Question 2Question →

What is the maximum integer value of xx that satisfies the inequality 3(2−3x)4−2(2x+5)3≥1−x2+76\frac{3(2 - 3x)}{4} - \frac{2(2x + 5)}{3} \geq \frac{1 - x}{2} + \frac{7}{6}?

Show answer & explanation

Answer: -2

Answer

The maximum integer value of xx that satisfies the inequality is −2-2.
Multiplying the inequality by the common denominator 12 and simplifying yields the inequality −37x≥42-37x \ge 42. Dividing by −37-37 requires reversing the inequality sign, which gives x≤−4237x \le -\frac{42}{37}. The value of −4237-\frac{42}{37} is approximately −1.135-1.135. The largest integer less than or equal to −1.135-1.135 is −2-2.

Step-by-Step Solution

1
Multiply both sides of the inequality by the least common multiple of the denominators (12).
9(2−3x)−8(2x+5)≥6(1−x)+149(2 - 3x) - 8(2x + 5) \geq 6(1 - x) + 14
This eliminates the fractions and simplifies the algebraic manipulation.
2
Expand the terms on both sides of the inequality.
18−27x−16x−40≥6−6x+1418 - 27x - 16x - 40 \geq 6 - 6x + 14
Expanding the terms allows us to combine like terms.
3
Combine the constant and variable terms on each side.
−43x−22≥20−6x-43x - 22 \geq 20 - 6x
This simplifies the inequality to a standard linear form.
4
Add 6x6x and 2222 to both sides to isolate the variable term on the left.
−37x≥42-37x \geq 42
Grouping variable terms on one side and constant terms on the other prepares for the final division.
5
Divide both sides by −37-37 and reverse the direction of the inequality sign.
x≤−4237x \leq -\frac{42}{37}
Dividing an inequality by a negative number requires flipping the inequality sign.
6
Find the largest integer that is less than or equal to −4237-\frac{42}{37}.
−2-2
Since −4237≈−1.135-\frac{42}{37} \approx -1.135, the integers less than or equal to this value are −2,−3,−4,…-2, -3, -4, \dots, of which −2-2 is the greatest.

Key Concept

Solving multi-step linear inequalities with rational coefficients, applying the inequality sign-flip rule, and finding boundary integer conditions.

Alternative Method

Instead of clearing the fractions first, you can group all terms containing xx on one side and the constant terms on the other side by finding a common denominator for only the variables and only the constants. However, clearing the fractions first is generally less prone to errors.
Estimated Time:2m 0s
Question 3Question →

A logistics company determines that its daily operating cost, CC (in dollars), for a delivery truck satisfies the inequality a(2C−3)3+54≤C−a2−78\frac{a(2C - 3)}{3} + \frac{5}{4} \leq \frac{C - a}{2} - \frac{7}{8}, where aa is a constant regional fuel efficiency parameter such that a<−2a < -2. Which of the following represents the range of possible operating costs CC?

Show answer & explanation

Answer: C≥51−12a12−16aC \geq \frac{51 - 12a}{12 - 16a}

Answer

The range of possible operating costs is C≥51−12a12−16aC \geq \frac{51 - 12a}{12 - 16a}.
To solve the inequality, we first eliminate the denominators by multiplying the entire inequality by 24, resulting in 8a(2C−3)+30≤12(C−a)−218a(2C - 3) + 30 \leq 12(C - a) - 21. Expanding both sides and gathering all terms with CC on the left gives (16a−12)C≤12a−51(16a - 12)C \leq 12a - 51. Because a<−2a < -2, the coefficient 16a−1216a - 12 is negative. Dividing by a negative number reverses the inequality direction, giving C≥12a−5116a−12C \geq \frac{12a - 51}{16a - 12}. Multiplying the numerator and denominator by −1-1 yields the correct solution.

Step-by-Step Solution

1
Clear the denominators by multiplying all terms by the least common multiple of 3, 4, 2, and 8, which is 24.
8a(2C−3)+30≤12(C−a)−218a(2C - 3) + 30 \leq 12(C - a) - 21
Eliminating fractions simplifies the algebraic manipulation of the linear inequality.
2
Expand both sides of the inequality.
16aC−24a+30≤12C−12a−2116aC - 24a + 30 \leq 12C - 12a - 21
Distributing terms allows grouping the variable CC and the constants.
3
Isolate the terms containing CC on the left side and all other terms on the right side.
16aC−12C≤12a−5116aC - 12C \leq 12a - 51
Grouping like terms is necessary to solve for CC.
4
Factor out CC on the left side.
(16a−12)C≤12a−51(16a - 12)C \leq 12a - 51
This isolates the variable CC with a single coefficient.
5
Determine the sign of the coefficient (16a−12)(16a - 12) based on the condition a<−2a < -2.
Since a<−2a < -2, we have 16a<−3216a < -32, which implies 16a−12<−4416a - 12 < -44. Thus, the coefficient is negative.
Knowing whether the coefficient is positive or negative determines whether the inequality sign must flip upon division.
6
Divide both sides by (16a−12)(16a - 12) and reverse the inequality sign.
C≥12a−5116a−12C \geq \frac{12a - 51}{16a - 12}
Dividing an inequality by a negative number requires reversing the direction of the inequality sign.
7
Simplify the resulting fraction by multiplying the numerator and denominator by −1-1.
C≥51−12a12−16aC \geq \frac{51 - 12a}{12 - 16a}
This yields the simplified final expression matching the target choice.

Key Concept

Solving linear inequalities involving fractions and variable parameters, with strict application of the inequality sign-flip rule when dividing by a negative algebraic term.
Estimated Time:3m 0s
Question 4Question →

For what greatest integer value of yy is the inequality 9−2y≥169 - 2y \geq 16 true?

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Answer: -4

Answer

The greatest integer value of yy that satisfies the inequality is −4-4.
Subtracting 9 from both sides of 9−2y≥169 - 2y \geq 16 gives −2y≥7-2y \geq 7. Dividing both sides by −2-2 and reversing the inequality sign yields y≤−3.5y \leq -3.5. The greatest integer less than or equal to −3.5-3.5 is −4-4.

Step-by-Step Solution

1
Subtract 9 from both sides of the inequality.
−2y≥7-2y \geq 7
This isolates the term containing yy on the left side.
2
Divide both sides of the inequality by −2-2 and reverse the direction of the inequality sign.
y≤−3.5y \leq -3.5
Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.
3
Identify the greatest integer that is less than or equal to −3.5-3.5.
−4-4
The value of yy must be less than or equal to −3.5-3.5. The integers satisfying this condition are −4,−5,−6,…-4, -5, -6, \dots, and the greatest of these is −4-4.

Key Concept

Solving linear inequalities by isolating the variable and reversing the inequality sign when dividing by a negative number.
Question 5Question →

A manufacturing company determines that its weekly profit from producing xx batches of a product is constrained by resource availability. To meet these resource constraints, the number of batches xx must satisfy the inequality:

5(6−x)3−3(x+2)4>2x−11\frac{5(6 - x)}{3} - \frac{3(x + 2)}{4} > 2x - 11

What is the greatest number of whole batches the company can produce while satisfying this constraint?

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Answer: 4

Answer

The greatest number of whole batches the company can produce is 4.
Solving the inequality yields x<23453≈4.415x < \frac{234}{53} \approx 4.415. The greatest integer value that satisfies this condition is 4.

Step-by-Step Solution

1
Multiply both sides of the inequality by the least common multiple of the denominators, which is 12.
20(6−x)−9(x+2)>24x−13220(6 - x) - 9(x + 2) > 24x - 132
Multiplying by 12 eliminates the fractions, making the inequality easier to solve.
2
Distribute the constants on the left side of the inequality.
120−20x−9x−18>24x−132120 - 20x - 9x - 18 > 24x - 132
Distributing 20 to (6−x)(6 - x) yields 120−20x120 - 20x, and distributing −9-9 to (x+2)(x + 2) yields −9x−18-9x - 18.
3
Combine like terms on the left side of the inequality.
102−29x>24x−132102 - 29x > 24x - 132
Combining 120−18120 - 18 gives 102102, and combining −20x−9x-20x - 9x gives −29x-29x.
4
Subtract 24x24x from both sides to group the variable terms on the left side.
102−53x>−132102 - 53x > -132
This groups all terms containing the variable xx on one side of the inequality.
5
Subtract 102 from both sides to isolate the variable term.
−53x>−234-53x > -234
This isolates the term containing xx on the left side of the inequality.
6
Divide both sides by −53-53 and reverse the inequality sign.
x<23453x < \frac{234}{53}
Dividing by a negative number requires reversing the direction of the inequality sign.
7
Evaluate the fraction as a decimal and determine the greatest integer value of xx that satisfies the inequality.
x<4.415x < 4.415, which means the greatest integer is 4.
Since the company must produce a whole number of batches, we find the largest integer less than 4.415.

Key Concept

Solving multi-step linear inequalities involving fractional coefficients, distributing negative numbers, and reversing the inequality sign when multiplying or dividing by a negative number.

Alternative Method

Instead of solving algebraically, you can test integer values for xx directly in the inequality. Testing x=4x = 4 gives 103−184=3.33−4.5=−1.17\frac{10}{3} - \frac{18}{4} = 3.33 - 4.5 = -1.17, which is greater than 2(4)−11=−32(4) - 11 = -3 (True). Testing x=5x = 5 gives 53−214=1.67−5.25=−3.58\frac{5}{3} - \frac{21}{4} = 1.67 - 5.25 = -3.58, which is not greater than 2(5)−11=−12(5) - 11 = -1 (False). This confirms 4 is the largest integer satisfying the inequality.
Estimated Time:2m 30s
Question 6Question →

For a real number xx, 12\frac{1}{2} minus 23\frac{2}{3} of xx is greater than 56\frac{5}{6}. Which of the following is the complete set of solutions for xx?

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Answer: x<−12x < -\frac{1}{2}

Answer

x<−12x < -\frac{1}{2}
Subtracting 12\frac{1}{2} from both sides of the inequality 12−23x>56\frac{1}{2} - \frac{2}{3}x > \frac{5}{6} gives −23x>26-\frac{2}{3}x > \frac{2}{6}, which simplifies to −23x>13-\frac{2}{3}x > \frac{1}{3}. Multiplying both sides by the negative fraction −32-\frac{3}{2} isolates xx on the left and reverses the inequality sign from greater than (>>) to less than (<<). Performing the multiplication on the right yields 13⋅(−32)=−12\frac{1}{3} \cdot \left(-\frac{3}{2}\right) = -\frac{1}{2}. Thus, the solution set is x<−12x < -\frac{1}{2}.

Step-by-Step Solution

1
Translate the verbal description into an algebraic inequality.
12−23x>56\frac{1}{2} - \frac{2}{3}x > \frac{5}{6}
The phrase '12\frac{1}{2} minus 23\frac{2}{3} of xx' represents the expression 12−23x\frac{1}{2} - \frac{2}{3}x, and 'is greater than' translates to the inequality symbol >>.
2
Subtract 12\frac{1}{2} from both sides of the inequality.
−23x>13-\frac{2}{3}x > \frac{1}{3}
To isolate the variable term on the left, we subtract 12\frac{1}{2} (which is equivalent to 36\frac{3}{6}) from 56\frac{5}{6} to get 26=13\frac{2}{6} = \frac{1}{3}.
3
Multiply both sides of the inequality by −32-\frac{3}{2} and reverse the inequality sign.
x<−12x < -\frac{1}{2}
Multiplying both sides of an inequality by a negative number requires reversing the direction of the inequality sign from >> to << to keep the inequality true.

Key Concept

Solving multi-step linear inequalities involving multiplication or division by a negative number.
Question 7Question →

When 55 is subtracted from 22 times a number xx, the result is at least 99. Which of the following inequalities represents all possible values of xx?

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Answer: x≥7x \geq 7

Answer

The inequality x≥7x \geq 7 represents all possible values of xx.
The verbal statement translates to the inequality 2x−5≥92x - 5 \geq 9. Adding 55 to both sides gives 2x≥142x \geq 14, and dividing both sides by the positive number 22 results in x≥7x \geq 7. Since we divide by a positive number, the direction of the inequality does not change.

Step-by-Step Solution

1
Translate the verbal phrase into an algebraic inequality.
2x−5≥92x - 5 \geq 9
'5 subtracted from 2 times a number xx' translates to 2x−52x - 5, and 'at least 9' means greater than or equal to 9.
2
Add 5 to both sides of the inequality to isolate the variable term.
2x≥142x \geq 14
Adding 5 to both sides maintains the inequality and simplifies the left side.
3
Divide both sides by 2.
x≥7x \geq 7
Dividing by a positive number does not change the direction of the inequality sign.

Key Concept

Solving linear inequalities by translating verbal statements into algebraic forms and applying inverse operations.
Question 8Question →

What is the smallest integer value of xx that satisfies the inequality 11−3x<211 - 3x < 2?

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Answer: 4

Answer

The smallest integer value of xx that satisfies the inequality is 4.
Solving the inequality 11−3x<211 - 3x < 2 leads to −3x<−9-3x < -9. Dividing by −3-3 and reversing the inequality sign gives x>3x > 3. The smallest integer that is strictly greater than 3 is 4.

Step-by-Step Solution

1
Isolate the variable term on one side of the inequality by subtracting 11 from both sides.
−3x<−9-3x < -9
Subtracting 11 from both sides of the inequality keeps the relationship balanced.
2
Divide both sides of the inequality by the coefficient of xx, which is −3-3.
x>3x > 3
Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.
3
Determine the smallest integer that is strictly greater than 3.
4
Since the inequality is strict (x>3x > 3), 3 is not included in the solution set. The smallest integer greater than 3 is 4.

Key Concept

Solving linear inequalities and reversing the inequality sign when dividing by a negative number.
Question 9Question →

For all real values of xx, which of the following inequalities represents the complete solution set to the inequality 5−2x3−3x−14≤x+22\frac{5 - 2x}{3} - \frac{3x - 1}{4} \leq \frac{x + 2}{2}?

Show answer & explanation

Answer: x≥1123x \geq \frac{11}{23}

Answer

The complete solution set is the set of all real numbers greater than or equal to 11/23.
The correct answer is found by clearing the denominators with the least common multiple of 12, carefully expanding the terms to get 20−8x−9x+3≤6x+1220 - 8x - 9x + 3 \leq 6x + 12, simplifying to 23−17x≤6x+1223 - 17x \leq 6x + 12, grouping terms to get −23x≤−11-23x \leq -11, and dividing by −23-23 which flips the sign to yield all real values greater than or equal to 11/23.

Step-by-Step Solution

1
Multiply all terms of the inequality by the least common multiple of the denominators (3, 4, and 2), which is 12, to clear the fractions.
4(5−2x)−3(3x−1)≤6(x+2)4(5 - 2x) - 3(3x - 1) \leq 6(x + 2)
Multiplying by a positive number allows us to eliminate denominators without changing the direction of the inequality.
2
Distribute the coefficients on both sides of the inequality, paying close attention to the distribution of the negative sign over the second term.
20−8x−9x+3≤6x+1220 - 8x - 9x + 3 \leq 6x + 12
Distributing −3-3 to both 3x3x and −1-1 yields −9x-9x and +3+3 respectively.
3
Combine the constant terms and the variable terms on the left side of the inequality.
23−17x≤6x+1223 - 17x \leq 6x + 12
Simplifying the expressions on each side makes the inequality easier to isolate.
4
Isolate the variable terms on the left and the constant terms on the right by subtracting 6x6x and 23 from both sides.
−23x≤−11-23x \leq -11
Grouping like terms together is necessary to solve for the variable.
5
Divide both sides of the inequality by −23-23 and reverse the direction of the inequality sign.
x≥1123x \geq \frac{11}{23}
Dividing both sides of an inequality by a negative number reverses the direction of the inequality sign from ≤\leq to ≥\geq.

Key Concept

Solving linear inequalities involving fractions and distributing negative coefficients, specifically applying the rule that multiplying or dividing by a negative number reverses the inequality direction.

Alternative Method

Instead of clearing the fractions first, write each fraction as separate terms: 53−23x−34x+14≤12x+1\frac{5}{3} - \frac{2}{3}x - \frac{3}{4}x + \frac{1}{4} \leq \frac{1}{2}x + 1. Then, collect the constant terms on one side and the variable terms on the other side using decimal or fractional conversions, and isolate the variable.
Estimated Time:2m 0s
Question 10Question →

What is the greatest integer value of xx that satisfies the inequality 2−5x3−x−42≥3\frac{2 - 5x}{3} - \frac{x - 4}{2} \geq 3?

Show answer & explanation

Answer: -1

Answer

The greatest integer value of xx that satisfies the inequality is -1.
The correct answer is -1 because solving the inequality leads to x≤−213x \leq -\frac{2}{13}. Since −213-\frac{2}{13} is approximately −0.154-0.154, the set of integers satisfying the inequality is {−1,−2,−3,… }\{-1, -2, -3, \dots\}. The greatest integer in this set is -1.

Step-by-Step Solution

1
Multiply the entire inequality by the least common multiple of the denominators (6) to eliminate the fractions.
2(2−5x)−3(x−4)≥182(2 - 5x) - 3(x - 4) \geq 18
Multiplying by a positive number clears the fractions without changing the direction of the inequality.
2
Distribute the coefficients and combine like terms on the left side of the inequality.
16−13x≥1816 - 13x \geq 18
Simplifying the expressions on each side makes it easier to isolate the variable.
3
Subtract 16 from both sides to isolate the term with the variable xx.
−13x≥2-13x \geq 2
Moving the constant terms to one side prepares the inequality for division.
4
Divide both sides by -13 and reverse the direction of the inequality sign.
x≤−213x \leq -\frac{2}{13}
Dividing by a negative number requires flipping the inequality sign to maintain a true statement.
5
Determine the greatest integer that is less than or equal to −213-\frac{2}{13}.
-1
Since −213≈−0.154-\frac{2}{13} \approx -0.154, the largest integer that is less than or equal to this value is -1.

Key Concept

Solving multi-step linear inequalities, including clearing fractional coefficients and reversing the inequality sign when multiplying or dividing by a negative number.
Question 11Question →

What is the maximum integer value of kk that satisfies the inequality 8−5k>288 - 5k > 28?

Show answer & explanation

Answer: -5

Answer

The maximum integer value that satisfies the inequality is −5-5.
Subtracting 8 from both sides of the inequality 8−5k>288 - 5k > 28 yields −5k>20-5k > 20. Dividing both sides of the inequality by −5-5 and reversing the inequality sign results in k<−4k < -4. The largest integer strictly less than −4-4 is −5-5.

Step-by-Step Solution

1
Subtract 8 from both sides of the inequality.
−5k>20-5k > 20
To isolate the term with the variable on the left side of the inequality.
2
Divide both sides by −5-5 and reverse the inequality sign.
k<−4k < -4
Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality symbol.
3
Determine the largest integer strictly less than −4-4.
−5-5
Because the inequality is strict (<<), the value of kk cannot be equal to −4-4. The greatest integer less than −4-4 is −5-5.

Key Concept

Solving linear inequalities and reversing the inequality sign when dividing by a negative number.
Question 12Question →

Which of the following inequalities represents the complete set of real values of xx that satisfy the inequality 15≤3−4x15 \leq 3 - 4x?

Show answer & explanation

Answer: x≤−3x \leq -3

Answer

The inequality x≤−3x \leq -3
Subtracting 3 from both sides of the inequality 15≤3−4x15 \leq 3 - 4x yields 12≤−4x12 \leq -4x. Dividing both sides by −4-4 and reversing the inequality sign results in −3≥x-3 \geq x, which is equivalent to x≤−3x \leq -3.

Step-by-Step Solution

1
Subtract 3 from both sides of the inequality.
12≤−4x12 \leq -4x
This isolates the variable term on the right side of the inequality.
2
Divide both sides of the inequality by −4-4 and reverse the inequality sign.
−3≥x-3 \geq x
Reversing the inequality sign is required whenever both sides of an inequality are multiplied or divided by a negative number.
3
Rewrite the inequality to place the variable on the left side.
x≤−3x \leq -3
Reorganizing the inequality with xx on the left side is the standard format for representing the solution set.

Key Concept

Solving linear inequalities by isolating the variable and reversing the inequality sign when dividing by a negative number.
Question 13Question →

A certain relationship between a number xx and other values is described as follows: one-fourth of the difference when 3x3x is subtracted from 22, decreased by one-third of the sum of xx and 33, is strictly greater than the difference when xx is subtracted from 11. Which of the following inequalities represents the complete set of all possible values of xx?

Show answer & explanation

Answer: x<−18x < -18

Answer

x<−18x < -18
To find the correct solution set, translate the word problem into the inequality 2−3x4−x+33>1−x\frac{2 - 3x}{4} - \frac{x + 3}{3} > 1 - x. First, multiply all terms by the least common denominator, 1212, to clear the fractions, giving 3(2−3x)−4(x+3)>12(1−x)3(2 - 3x) - 4(x + 3) > 12(1 - x). Distributing the constants yields 6−9x−4x−12>12−12x6 - 9x - 4x - 12 > 12 - 12x. Combining like terms on the left side simplifies the expression to −13x−6>12−12x-13x - 6 > 12 - 12x. Adding 12x12x to both sides results in −x−6>12-x - 6 > 12. Adding 66 to both sides gives −x>18-x > 18. Finally, dividing by −1-1 and reversing the inequality sign results in the solution set x<−18x < -18.

Step-by-Step Solution

1
Translate the verbal description into an algebraic inequality.
2−3x4−x+33>1−x\frac{2 - 3x}{4} - \frac{x + 3}{3} > 1 - x
To represent the relationships described in the word problem mathematically.
2
Multiply the entire inequality by the least common denominator, 12, to clear the fractions.
3(2−3x)−4(x+3)>12(1−x)3(2 - 3x) - 4(x + 3) > 12(1 - x)
Clearing denominators makes it easier to combine like terms and solve for xx.
3
Distribute the constants on both sides.
6−9x−4x−12>12−12x6 - 9x - 4x - 12 > 12 - 12x
To remove parentheses and separate individual terms for simplification.
4
Combine like terms on the left side of the inequality.
−13x−6>12−12x-13x - 6 > 12 - 12x
Simplifying the expressions on each side makes the inequality easier to solve.
5
Add 12x12x and 66 to both sides of the inequality to isolate variables on one side.
−x>18-x > 18
To group variable terms on the left and constant terms on the right.
6
Divide both sides by −1-1 and reverse the inequality sign.
x<−18x < -18
Dividing or multiplying an inequality by a negative number requires reversing the inequality sign to maintain a true statement.

Key Concept

Solving multi-step linear inequalities with rational terms and variable terms on both sides, including reversing the inequality sign when dividing by a negative number.

Alternative Method

Instead of clearing fractions first, you can distribute the division to each term (e.g., 24−3x4−x3−33>1−x\frac{2}{4} - \frac{3x}{4} - \frac{x}{3} - \frac{3}{3} > 1 - x), group the xx terms on one side and constants on the other using fraction arithmetic, and then solve for xx. However, clearing fractions with the LCD is generally faster and less prone to arithmetic errors.
Estimated Time:2m 0s
Question 14Question →

A manufacturing company produces two types of metal alloys. The production cost, in dollars per ton, of Alloy A is modeled by the expression 800−15(x−5)800 - 15(x - 5), where xx is the amount of stabilizer added in kilograms. The production cost of Alloy B, in dollars per ton, is modeled by the expression 520+5(x+3)520 + 5(x + 3). If the company requires the production cost of Alloy A to be strictly less than the production cost of Alloy B, what is the minimum integer amount of stabilizer xx, in kilograms, that must be added?

Show answer & explanation

Answer: 18

Answer

The minimum integer amount of stabilizer that must be added is 1818 kg.
The inequality representing the condition is 800−15(x−5)<520+5(x+3)800 - 15(x - 5) < 520 + 5(x + 3). Expanding both sides gives 875−15x<535+5x875 - 15x < 535 + 5x. Isolating xx yields −20x<−340-20x < -340. Dividing by −20-20 and flipping the inequality sign results in x>17x > 17. The minimum integer value that is strictly greater than 1717 is 1818.

Step-by-Step Solution

1
Set up the linear inequality using the cost expressions for Alloy A and Alloy B.
800−15(x−5)<520+5(x+3)800 - 15(x - 5) < 520 + 5(x + 3)
The cost of Alloy A must be strictly less than the cost of Alloy B.
2
Distribute and combine like terms to simplify both sides of the inequality.
875−15x<535+5x875 - 15x < 535 + 5x
Simplifying the expressions makes it easier to isolate the variable.
3
Isolate the variable term on one side of the inequality.
−20x<−340-20x < -340
Subtracting 5x5x and 875875 from both sides moves all variable terms to the left and constant terms to the right.
4
Divide both sides by the negative coefficient −20-20 and reverse the inequality sign.
x>17x > 17
Dividing an inequality by a negative number requires reversing the direction of the inequality sign.
5
Identify the smallest integer that satisfies the inequality.
1818
The solution requires a strict inequality x>17x > 17, so the smallest integer value that is strictly greater than 1717 is 1818.

Key Concept

Solving multi-step linear inequalities with variables on both sides, including reversing the inequality sign when dividing by a negative number.
Question 15Question →

A manufacturer calculates the net monthly operating profit PP (in thousands of dollars) using the expression 8−3x2\frac{8 - 3x}{2}, where xx is the number of maintenance hours scheduled. The auxiliary support costs are modeled by the expression 2x−13\frac{2x - 1}{3} thousand dollars. For the plant to be considered efficient, the net operating profit minus the auxiliary support costs must be at least 33 thousand dollars. Which of the following inequalities represents the range of maintenance hours, xx, that satisfy this efficiency requirement?

Show answer & explanation

Answer: x≤813x \leq \frac{8}{13}

Answer

The range of maintenance hours that satisfy the efficiency requirement is x≤813x \leq \frac{8}{13}.
First, translate the word problem into a mathematical inequality. The net monthly operating profit minus the auxiliary support costs must be at least 3, which translates to: 8−3x2−2x−13≥3\frac{8 - 3x}{2} - \frac{2x - 1}{3} \geq 3 To solve this inequality, find the least common denominator (LCD) of 2 and 3, which is 6. Multiply every term in the inequality by 6: 3(8−3x)−2(2x−1)≥183(8 - 3x) - 2(2x - 1) \geq 18 Distribute the constants on the left side, paying careful attention to distribute the negative sign: 24−9x−4x+2≥1824 - 9x - 4x + 2 \geq 18 Combine like terms: 26−13x≥1826 - 13x \geq 18 Subtract 26 from both sides: −13x≥−8-13x \geq -8 Divide both sides by -13. Because we are dividing by a negative number, the direction of the inequality sign must be reversed: x≤813x \leq \frac{8}{13} Thus, the option stating x≤813x \leq \frac{8}{13} is correct.

Step-by-Step Solution

1
Translate the word problem into a mathematical inequality.
8−3x2−2x−13≥3\frac{8 - 3x}{2} - \frac{2x - 1}{3} \geq 3
The net operating profit minus the auxiliary support costs must be at least 3, which translates to a subtraction operation set greater than or equal to 3.
2
Multiply all terms in the inequality by the least common denominator (LCD) of 2 and 3, which is 6.
3(8−3x)−2(2x−1)≥183(8 - 3x) - 2(2x - 1) \geq 18
Multiplying by the LCD eliminates the denominators and simplifies the equation for algebraic manipulation.
3
Distribute the constants on the left side of the inequality.
24−9x−4x+2≥1824 - 9x - 4x + 2 \geq 18
Removing the parentheses is required to combine like terms. Be careful to distribute the negative sign: −2×−1=+2-2 \times -1 = +2.
4
Combine like terms on the left side of the inequality.
26−13x≥1826 - 13x \geq 18
Simplifying the left-hand side reduces the inequality to a standard two-step linear inequality.
5
Subtract 26 from both sides of the inequality to isolate the variable term.
−13x≥−8-13x \geq -8
This isolates the variable term −13x-13x on the left side.
6
Divide both sides by -13 and reverse the direction of the inequality sign.
x≤813x \leq \frac{8}{13}
Dividing both sides of an inequality by a negative number requires reversing the inequality sign to maintain equivalence.

Key Concept

Solving multi-step linear inequalities, including fraction clearance and reversing the inequality sign when dividing by a negative number.
Question 16Question →

What is the greatest integer value of xx that satisfies the inequality 3−2x>103 - 2x > 10?

Show answer & explanation

Answer: -4

Answer

The correct answer is −4-4.
Subtracting 3 from both sides of 3−2x>103 - 2x > 10 gives −2x>7-2x > 7. When dividing both sides by −2-2, the inequality sign must be flipped, yielding x<−3.5x < -3.5. The greatest integer less than −3.5-3.5 is −4-4.

Step-by-Step Solution

1
Subtract 3 from both sides of the inequality to isolate the variable term.
−2x>7-2x > 7
Subtracting 3 from both sides keeps the inequality balanced while moving the constant term to the right side.
2
Divide both sides by −2-2 and reverse the inequality sign.
x<−3.5x < -3.5
Dividing or multiplying an inequality by a negative number requires reversing the direction of the inequality sign to maintain a true statement.
3
Identify the greatest integer that satisfies the inequality.
−4-4
The integers that are strictly less than −3.5-3.5 are −4,−5,−6,…-4, -5, -6, \dots. The largest (greatest) of these integers is −4-4.

Key Concept

Solving linear inequalities by applying the sign-reversal rule when dividing by a negative number and identifying integer boundary values.
Question 17Question →

Which inequality represents all real values of xx for which the inequality 5−2x3≥x−62\frac{5 - 2x}{3} \geq \frac{x - 6}{2} is true?

Show answer & explanation

Answer: x≤4x \leq 4

Answer

The correct inequality is x≤4x \leq 4.
The correct inequality is x≤4x \leq 4. Multiplying both sides by the least common multiple, 66, yields the inequality 2(5−2x)≥3(x−6)2(5 - 2x) \geq 3(x - 6). Distributing the coefficients results in 10−4x≥3x−1810 - 4x \geq 3x - 18. Gathering the variable terms by subtracting 3x3x gives 10−7x≥−1810 - 7x \geq -18. Subtracting 1010 from both sides results in −7x≥−28-7x \geq -28. Dividing both sides by −7-7 and reversing the inequality sign results in the final solution x≤4x \leq 4.

Step-by-Step Solution

1
Multiply both sides of the inequality by 66 (the least common multiple of 22 and 33) to eliminate the fractions.
2(5−2x)≥3(x−6)2(5 - 2x) \geq 3(x - 6)
Eliminating denominators simplifies the linear inequality for solving.
2
Distribute the constants on both sides.
10−4x≥3x−1810 - 4x \geq 3x - 18
Expanding the terms allows combining like terms next.
3
Subtract 3x3x from both sides of the inequality.
10−7x≥−1810 - 7x \geq -18
Grouping all variable terms on one side of the inequality.
4
Subtract 1010 from both sides of the inequality.
−7x≥−28-7x \geq -28
Isolating the variable term on the left side.
5
Divide both sides by −7-7 and reverse the inequality sign.
x≤4x \leq 4
Dividing or multiplying both sides of an inequality by a negative number requires reversing the inequality sign direction.

Key Concept

Solving linear inequalities by clearing denominators and applying the sign-reversal rule when dividing by a negative number.
Estimated Time:1m 15s
Question 18Question →

What is the smallest integer value of yy that satisfies the inequality 5(2−y)<3(y−6)5(2 - y) < 3(y - 6)?

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Answer: 4

Answer

The smallest integer value of yy that satisfies the inequality is 44.
Evaluating the inequality leads to y>3.5y > 3.5. The smallest integer greater than 3.53.5 is 44. Substituting y=4y = 4 into the original inequality gives 5(2−4)<3(4−6)  ⟹  −10<−65(2 - 4) < 3(4 - 6) \implies -10 < -6, which is true. Substituting the next smallest integer, 33, gives −5<−9-5 < -9, which is false.

Step-by-Step Solution

1
Distribute the coefficients to the terms inside the parentheses on both sides.
10−5y<3y−1810 - 5y < 3y - 18
To clear the parentheses and simplify the terms.
2
Subtract 3y3y and 1010 from both sides of the inequality to group the variable terms on one side and constants on the other.
−8y<−28-8y < -28
To isolate the variable term.
3
Divide both sides by −8-8 and reverse the inequality sign because of division by a negative number.
y>3.5y > 3.5
To solve the inequality for yy.
4
Identify the smallest integer that satisfies the inequality y>3.5y > 3.5.
44
To find the smallest integer value greater than 3.53.5.

Key Concept

Solving multi-step linear inequalities involving distribution and division by a negative number.

Alternative Method

Instead of subtracting variables to the left, we can add 5y5y to both sides to keep the variable coefficient positive: 10−5y<3y−18  ⟹  10<8y−18  ⟹  28<8y  ⟹  y>3.510 - 5y < 3y - 18 \implies 10 < 8y - 18 \implies 28 < 8y \implies y > 3.5. This avoids the need to divide by a negative number and flip the sign, reducing the risk of a sign-flip error.
Estimated Time:1m 0s
Question 19Question →

A student translates a word problem into an inequality. The problem states: "One-fourth of the difference of xx and 33 is greater than the sum of 11 and one-third of the quantity 2x2x minus 11." Which of the following inequalities represents the correct set of all real values of xx that satisfy this condition?

Show answer & explanation

Answer: x<−175x < -\frac{17}{5}

Answer

The set of all real values of xx satisfying the condition is x<−175x < -\frac{17}{5}.
The correct inequality is obtained by translating the word problem statement as x−34>1+2x−13\frac{x - 3}{4} > 1 + \frac{2x - 1}{3}. Multiplying the entire inequality by 1212 yields 3(x−3)>12+4(2x−1)3(x - 3) > 12 + 4(2x - 1). Distributing the coefficients results in 3x−9>12+8x−43x - 9 > 12 + 8x - 4, which simplifies to 3x−9>8x+83x - 9 > 8x + 8. Moving all terms containing xx to the left and constants to the right gives −5x>17-5x > 17. Dividing both sides by −5-5 and reversing the inequality sign results in the solution x<−175x < -\frac{17}{5}.

Step-by-Step Solution

1
Translate the verbal description into an algebraic inequality.
x−34>1+2x−13\frac{x - 3}{4} > 1 + \frac{2x - 1}{3}
The phrase "one-fourth of the difference of xx and 33" translates to x−34\frac{x - 3}{4}, and "the sum of 11 and one-third of the quantity 2x2x minus 11" translates to 1+2x−131 + \frac{2x - 1}{3}.
2
Multiply all terms by the least common multiple of the denominators, which is 1212, to clear the fractions.
3(x−3)>12+4(2x−1)3(x - 3) > 12 + 4(2x - 1)
Multiplying both sides of an inequality by a positive number maintains the direction of the inequality sign while eliminating fractional coefficients.
3
Expand both sides of the inequality by distributing the coefficients.
3x−9>12+8x−43x - 9 > 12 + 8x - 4
Distribution allows the variable and constant terms to be separated and simplified.
4
Simplify the constants on the right side and move all variable terms to one side and constants to the other.
−5x>17-5x > 17
Subtracting 8x8x from both sides and adding 99 to both sides isolates the variable term on the left side of the inequality.
5
Divide both sides by −5-5 and reverse the inequality sign.
x<−175x < -\frac{17}{5}
Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.

Key Concept

Solving linear inequalities by translating word problems and applying the sign-flip rule when dividing by a negative number.
Estimated Time:1m 30s
Question 20Question →

What is the greatest integer value of kk that satisfies the inequality 8−3(2k−5)≥4(k+6)8 - 3(2k - 5) \geq 4(k + 6)?

Show answer & explanation

Answer: -1

Answer

The greatest integer value of kk that satisfies the inequality is −1-1.
Solving the inequality step-by-step yields k≤−0.1k \leq -0.1. The greatest integer less than or equal to −0.1-0.1 is −1-1.

Step-by-Step Solution

1
Distribute the coefficients to remove parentheses
8−6k+15≥4k+248 - 6k + 15 \geq 4k + 24
Expanding the terms makes it possible to combine like terms on each side of the inequality.
2
Combine like terms on the left side
23−6k≥4k+2423 - 6k \geq 4k + 24
Simplifying the constant values on the left side (8+15=238 + 15 = 23) simplifies the expression.
3
Subtract 4k4k from both sides
23−10k≥2423 - 10k \geq 24
This groups all the variable terms on the left-hand side.
4
Subtract 2323 from both sides
−10k≥1-10k \geq 1
This isolates the variable term on the left-hand side.
5
Divide by −10-10 and flip the inequality sign
k≤−0.1k \leq -0.1
Dividing both sides by a negative number requires reversing the direction of the inequality sign.
6
Identify the greatest integer satisfying the inequality
k=−1k = -1
The largest integer that is less than or equal to −0.1-0.1 is −1-1.

Key Concept

Solving linear inequalities and applying the sign-flip rule when dividing by a negative number.
Estimated Time:1m 30s
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Solving Linear Inequalities Practice Questions — ACT | Examkin