Examkin

Solving One-Variable Inequalities

24 questions

All types24Multiple choice13True / False7Numeric4
Question 1Question →

Find the greatest integer value of xx that makes the following inequality true:

23(6x−9)−4(2x+1)>14(8x−20)−3x \frac{2}{3}(6x - 9) - 4(2x + 1) > \frac{1}{4}(8x - 20) - 3x
Show answer & explanation

Answer: -2

Answer

The greatest integer value that satisfies the inequality is -2.
Expanding both sides yields −4x−10>−x−5-4x - 10 > -x - 5. Isolating xx on one side gives −3x>5-3x > 5. Dividing by -3 requires flipping the inequality symbol, resulting in x<−53x < -\frac{5}{3} (or x<−1.67x < -1.67). The greatest integer strictly less than -1.67 is -2.

Step-by-Step Solution

1
Distribute the constants and fractions into the parentheses on both sides.
4x−6−8x−4>2x−5−3x4x - 6 - 8x - 4 > 2x - 5 - 3x
Applying the distributive property removes the parentheses so like terms can be combined.
2
Combine like terms on each side of the inequality.
−4x−10>−x−5-4x - 10 > -x - 5
Simplifying both expressions makes it easier to isolate the variable.
3
Add xx to both sides and add 1010 to both sides to separate the variable terms from the constant terms.
−3x>5-3x > 5
This isolates the xx term on one side of the inequality.
4
Divide both sides by -3 and flip the inequality sign.
x<−53x < -\frac{5}{3}
When multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed.
5
Identify the greatest integer less than −53-\frac{5}{3}.
Since −53≈−1.67-\frac{5}{3} \approx -1.67, the integers strictly less than this are −2,−3,−4,…-2, -3, -4, \dots The greatest of these is −2-2.
The question specifically asks for the greatest integer value that makes the inequality true.

Key Concept

Solving complex linear inequalities involves applying the distributive property, combining like terms, and reversing the inequality symbol when dividing or multiplying by a negative number.
Question 2Question →

Evaluate whether the following statement is true or false:

The only positive integer solutions to the linear inequality −35(103x−15)+4(x−2)≥−52(−2x+4)+2-\frac{3}{5}\left(\frac{10}{3}x - 15\right) + 4(x - 2) \geq -\frac{5}{2}(-2x + 4) + 2 are 1, 2, and 3.
Show answer & explanation

Answer: True

Answer

The statement is True.
By successfully distributing the fractional negative coefficients and combining like terms, the inequality simplifies to −3x≥−9-3x \geq -9. Dividing by −3-3 requires reversing the inequality to x≤3x \leq 3. The positive integers in this solution set are exactly 1, 2, and 3, meaning the statement is factually correct.

Step-by-Step Solution

1
Distribute the negative fractions and constants on the left side of the inequality.
(−2x+9)+(4x−8)(-2x + 9) + (4x - 8)
Multiplying −35-\frac{3}{5} by 103x\frac{10}{3}x gives −2x-2x, and −35-\frac{3}{5} by −15-15 gives +9+9. Multiplying 44 by (x−2)(x-2) gives 4x−84x - 8.
2
Distribute the negative fraction on the right side of the inequality.
(5x−10)+2(5x - 10) + 2
Multiplying −52-\frac{5}{2} by −2x-2x gives 5x5x, and −52-\frac{5}{2} by 44 gives −10-10.
3
Combine the like terms on both sides.
2x+1≥5x−82x + 1 \geq 5x - 8
On the left side, −2x+4x=2x-2x + 4x = 2x and 9−8=19 - 8 = 1. On the right side, −10+2=−8-10 + 2 = -8.
4
Isolate the variable xx by subtracting 5x5x and 11 from both sides.
−3x≥−9-3x \geq -9
Grouping all variables on one side and constants on the other prepares the equation for the final division.
5
Divide both sides by −3-3 and reverse the inequality symbol.
x≤3x \leq 3
Dividing an inequality by a negative number requires flipping the inequality direction. Since x≤3x \leq 3, the only integers greater than zero that satisfy this are 1, 2, and 3.

Key Concept

Solving complex linear inequalities by distributing negative fractions, combining like terms, and properly applying the inequality sign flip rule.

Alternative Method

You can verify the statement by plugging the boundary values into the original inequality. Substitute x=3x=3 (which evaluates to 7≥77 \geq 7, a true statement), x=4x=4 (which evaluates to 9≥129 \geq 12, a false statement), and recognize that since it is a linear relation, solutions are valid downwards. Zero is not a positive integer, leaving 1, 2, and 3 as the only valid positive integers.
Estimated Time:2m 30s
Question 3Question →

For the inequality below, which of the following expresses all possible values of yy?

2(3y−5)−3(y−4)≥6y−162(3y - 5) - 3(y - 4) \geq 6y - 16
Show answer & explanation

Answer: y≤6y \leq 6

Answer

The correct answer is y≤6y \leq 6.
The correct answer is derived by distributing the terms to get 6y−10−3y+12≥6y−166y - 10 - 3y + 12 \geq 6y - 16, which simplifies to 3y+2≥6y−163y + 2 \geq 6y - 16. Subtracting 6y6y from both sides and subtracting 2 from both sides yields −3y≥−18-3y \geq -18. Finally, dividing by -3 requires flipping the inequality symbol, resulting in y≤6y \leq 6.

Step-by-Step Solution

1
Distribute the constants on the left side of the inequality.
6y−10−3y+12≥6y−166y - 10 - 3y + 12 \geq 6y - 16
To clear the parentheses. Note that multiplying -3 by -4 results in a positive 12.
2
Combine the like terms on the left side.
3y+2≥6y−163y + 2 \geq 6y - 16
To simplify the expression before isolating the variable.
3
Move all variable terms to the left side and constant terms to the right side.
3y−6y≥−16−23y - 6y \geq -16 - 2, which simplifies to −3y≥−18-3y \geq -18
To group the unknown variable on one side of the inequality.
4
Divide both sides by -3.
y≤6y \leq 6
To solve for y. Whenever multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed.

Key Concept

Solving One-Variable Inequalities
Question 4Question →

For the linear inequality −34(8x−12)+2x>−3(x−4)−2x-\frac{3}{4}(8x - 12) + 2x > -3(x - 4) - 2x, the value x=3x = 3 is included in the solution set.

Show answer & explanation

Answer: False

Answer

False, because the solution set is x>3x > 3, which strictly excludes the boundary value of 3.
Simplifying both sides of the inequality yields −4x+9>−5x+12-4x + 9 > -5x + 12. Isolating xx provides the solution set x>3x > 3. Because the inequality is strictly 'greater than', it does not include the boundary value of 3, making the statement false.

Step-by-Step Solution

1
Distribute the negative fraction −34-\frac{3}{4} on the left side of the inequality.
−34(8x)−(−34)(12)+2x=−6x+9+2x-\frac{3}{4}(8x) - (-\frac{3}{4})(12) + 2x = -6x + 9 + 2x
To clear the parentheses and prepare the left side for combining like terms.
2
Distribute the negative integer −3-3 on the right side of the inequality.
−3(x)−(−3)(4)−2x=−3x+12−2x-3(x) - (-3)(4) - 2x = -3x + 12 - 2x
To clear the parentheses on the right side of the inequality.
3
Combine the like xx terms on both sides of the inequality.
Left side: −6x+2x+9=−4x+9-6x + 2x + 9 = -4x + 9. Right side: −3x−2x+12=−5x+12-3x - 2x + 12 = -5x + 12. The simplified inequality is −4x+9>−5x+12-4x + 9 > -5x + 12.
To simplify the expression before isolating the variable.
4
Isolate the variable xx by adding 5x5x and subtracting 99 on both sides.
−4x+5x>12−9-4x + 5x > 12 - 9, which simplifies to x>3x > 3.
To determine the final solution set for the inequality.
5
Evaluate whether x=3x = 3 satisfies the solution set x>3x > 3.
Since 33 is not greater than 33, the value is not included in the solution set.
To definitively determine if the original statement is true or false.

Key Concept

Solving multi-step linear inequalities and determining boundary value inclusion
Question 5Question →

Which of the following represents the correct solution set for the inequality below?

4(x−3)−3(2x+5)≥x−124(x - 3) - 3(2x + 5) \geq x - 12
Show answer & explanation

Answer: x≤−5x \leq -5

Answer

The correct solution is x≤−5x \leq -5.
The correct answer is derived by carefully distributing the values across the parentheses (yielding 4x−12−6x−15≥x−124x - 12 - 6x - 15 \geq x - 12), combining the like terms to form −2x−27≥x−12-2x - 27 \geq x - 12, and isolating xx to get −3x≥15-3x \geq 15. Crucially, dividing both sides by −3-3 requires reversing the direction of the inequality, which correctly results in x≤−5x \leq -5.

Step-by-Step Solution

1
Distribute the coefficients outside the parentheses into the terms inside.
4x−12−6x−15≥x−124x - 12 - 6x - 15 \geq x - 12
This removes the parentheses. Care must be taken to multiply -3 by +5 correctly to get -15.
2
Combine the like terms on the left side of the inequality.
−2x−27≥x−12-2x - 27 \geq x - 12
Simplifying both sides makes it easier to isolate the variable.
3
Move all terms containing xx to one side and all constant terms to the other side.
−3x≥15-3x \geq 15
Subtracting xx from both sides gives −3x-3x, and adding 2727 to both sides gives 1515.
4
Divide both sides by -3 and reverse the inequality symbol.
x≤−5x \leq -5
Whenever you multiply or divide an inequality by a negative number, the direction of the inequality sign must be flipped to maintain a true mathematical statement.

Key Concept

Solving multi-step linear inequalities involving distribution and the inequality sign flip rule.
Question 6Question →

What is the complete solution set for the given inequality?

12(−8x−10)−23(9x+15)≤−4(x−2)+2x+1 \frac{1}{2}(-8x - 10) - \frac{2}{3}(9x + 15) \leq -4(x - 2) + 2x + 1
Show answer & explanation

Answer: x≥−3x \geq -3

Answer

The correct solution is x≥−3x \geq -3.
By systematically distributing the fractions and negative integers on both sides, we simplify the expression to −10x−15≤−2x+9-10x - 15 \leq -2x + 9. Moving the variable terms to the left and constants to the right results in −8x≤24-8x \leq 24. Finally, dividing by −8-8 isolates xx and requires flipping the inequality symbol, providing the correct solution.

Step-by-Step Solution

1
Distribute the fractions into the terms inside the parentheses on the left side.
(−4x−5)−(6x+10)≤−4(x−2)+2x+1(-4x - 5) - (6x + 10) \leq -4(x - 2) + 2x + 1
To eliminate the parentheses and clear the fractions for easier manipulation.
2
Distribute the −4-4 on the right side of the inequality.
−4x−5−6x−10≤−4x+8+2x+1-4x - 5 - 6x - 10 \leq -4x + 8 + 2x + 1
To expand all terms fully before attempting to combine them.
3
Combine the like terms on both the left and right sides independently.
−10x−15≤−2x+9-10x - 15 \leq -2x + 9
To simplify the inequality into a standard two-step format.
4
Add 2x2x to both sides and add 1515 to both sides to group variable terms and constant terms.
−8x≤24-8x \leq 24
To isolate the xx term on one side of the inequality.
5
Divide both sides by −8-8 and reverse the direction of the inequality symbol.
x≥−3x \geq -3
Multiplying or dividing an inequality by a negative number requires flipping the inequality sign to maintain the mathematical truth.

Key Concept

Solving Complex Linear Inequalities
Question 7Question →

Which of the following represents the solution set for the inequality below?

−3(x−2)+5x>6x−10-3(x - 2) + 5x > 6x - 10
Show answer & explanation

Answer: x<4x < 4

Answer

The correct answer is x<4x < 4.
The correct solution is achieved by properly expanding the expression, isolating the variable xx, and correctly reversing the inequality symbol when dividing both sides by a negative number (−4-4).

Step-by-Step Solution

1
Distribute the −3-3 into the parentheses.
−3x+6+5x>6x−10-3x + 6 + 5x > 6x - 10
Expanding the terms allows for the combination of like variables.
2
Combine like terms on the left side of the inequality.
2x+6>6x−102x + 6 > 6x - 10
Simplifying the expression makes it easier to isolate the variable.
3
Subtract 6x6x from both sides of the inequality.
−4x+6>−10-4x + 6 > -10
This step groups all variable terms onto one side of the inequality.
4
Subtract 66 from both sides.
−4x>−16-4x > -16
This isolates the variable term completely.
5
Divide both sides by −4-4 and reverse the inequality sign.
x<4x < 4
Dividing or multiplying an inequality by a negative number requires flipping the inequality direction.

Key Concept

Solving one-variable inequalities requiring distribution and division by a negative coefficient.
Estimated Time:1m 0s
Question 8Question →

The value x=2x = 2 is a valid solution for the inequality 5−2(3x+1)<4x−175 - 2(3x + 1) < 4x - 17.

Show answer & explanation

Answer: False

Answer

False
Substituting x=2x = 2 into both sides of the inequality results in −9<−9-9 < -9. Because −9-9 is equal to −9-9 and not strictly less than it, the mathematical statement is false. Alternatively, fully solving the inequality yields x>2x > 2, which does not include 22 as a solution.

Step-by-Step Solution

1
Substitute x=2x = 2 into the left side of the inequality.
5−2(3(2)+1)=5−2(6+1)=5−2(7)=5−14=−95 - 2(3(2) + 1) = 5 - 2(6 + 1) = 5 - 2(7) = 5 - 14 = -9
To evaluate the numerical value of the left expression for the given test value.
2
Substitute x=2x = 2 into the right side of the inequality.
4(2)−17=8−17=−94(2) - 17 = 8 - 17 = -9
To evaluate the numerical value of the right expression for the given test value.
3
Compare the two evaluated sides using the strict inequality symbol (<<).
−9<−9-9 < -9
To determine if the resulting mathematical relationship is true.
4
Evaluate the truth of the final statement.
Since −9-9 is exactly equal to −9-9, it is not strictly less than −9-9. The statement is therefore false.
A strict inequality requires the left side value to be genuinely smaller than the right side value.

Key Concept

Solving One-Variable Inequalities
Question 9Question →

The solution set for the linear inequality 2(3x−5)−4x>8x−222(3x - 5) - 4x > 8x - 22 is x>2x > 2.

Show answer & explanation

Answer: False

Answer

The statement is false. The correct solution set is x<2x < 2.
The statement is false because solving the inequality gives x<2x < 2. The inequality symbol must be flipped when dividing the final term −6x>−12-6x > -12 by the negative coefficient −6-6.

Step-by-Step Solution

1
Distribute the 22 across the terms inside the parentheses.
6x−10−4x>8x−226x - 10 - 4x > 8x - 22
To eliminate the parentheses and prepare the terms for combining.
2
Combine like terms on the left side of the inequality.
2x−10>8x−222x - 10 > 8x - 22
To simplify the expression on the left side (6x−4x=2x6x - 4x = 2x).
3
Subtract 8x8x from both sides.
−6x−10>−22-6x - 10 > -22
To gather all the variable terms on one side of the inequality.
4
Add 1010 to both sides.
−6x>−12-6x > -12
To isolate the variable term completely.
5
Divide both sides by −6-6 and reverse the inequality sign.
x<2x < 2
Dividing an inequality by a negative number requires flipping the inequality symbol to maintain a mathematically true statement.

Key Concept

Solving multi-step linear inequalities and reversing the inequality symbol when dividing by a negative number.
Question 10Question →

Consider the following one-variable inequality:

12−3x≤2712 - 3x \leq 27

Which of the following represents the correct solution set for xx?

Show answer & explanation

Answer: x≥−5x \geq -5

Answer

x≥−5x \geq -5
To solve the inequality, first subtract 12 from both sides to get −3x≤15-3x \leq 15. Next, divide both sides by -3. Because the division involves a negative number, the direction of the inequality symbol must be reversed, resulting in the correct solution of x≥−5x \geq -5.

Step-by-Step Solution

1
Subtract 1212 from both sides of the inequality.
−3x≤15-3x \leq 15
This step isolates the term containing the variable on one side of the inequality.
2
Divide both sides by −3-3 and reverse the inequality symbol.
x≥−5x \geq -5
According to the rules of algebra, whenever you multiply or divide an inequality by a negative number, the direction of the inequality sign must be flipped.

Key Concept

Solving One-Variable Inequalities
Estimated Time:45s
Question 11Question →

The solution to the one-variable inequality −2x>10-2x > 10 is x>−5x > -5.

Show answer & explanation

Answer: False

Answer

False
The correct answer is False because isolating xx requires dividing by −2-2, which means the greater-than sign (>>) must flip to a less-than sign (<<). The actual mathematically correct solution is x<−5x < -5.

Step-by-Step Solution

1
Divide both sides of the inequality by −2-2.
−2x−2\frac{-2x}{-2} and 10−2\frac{10}{-2}
To isolate the variable xx on one side of the inequality.
2
Reverse the inequality symbol.
x<−5x < -5
When dividing or multiplying an inequality by a negative number, the direction of the inequality symbol must be flipped to maintain a mathematically true statement.

Key Concept

Solving One-Variable Inequalities
Question 12Question →

Multiplying both sides of the inequality −x3<5-\frac{x}{3} < 5 by −3-3 results in the equivalent inequality x>−15x > -15.

Show answer & explanation

Answer: True

Answer

True
The statement correctly applies the algebraic rule that multiplying an inequality by a negative number reverses the direction of the inequality sign, properly yielding x>−15x > -15.

Step-by-Step Solution

1
Identify the operation needed to isolate the variable xx.
Multiply both sides of the inequality −x3<5-\frac{x}{3} < 5 by −3-3.
The variable xx is being divided by −3-3, so multiplying by −3-3 is the inverse operation.
2
Perform the multiplication and apply the inequality division/multiplication rule.
−3×(−x3)>−3×5-3 \times (-\frac{x}{3}) > -3 \times 5, which simplifies to x>−15x > -15.
When multiplying or dividing an inequality by a negative number, the inequality symbol must be reversed to maintain a true mathematical statement.

Key Concept

When solving inequalities, multiplying or dividing both sides by a negative number requires flipping the inequality sign.
Question 13Question →

Which of the following describes the correct solution set for the inequality −2x−3≤9-2x - 3 \leq 9?

Show answer & explanation

Answer: x≥−6x \geq -6

Answer

The correct solution is x≥−6x \geq -6.
To solve the inequality, first isolate the variable term by adding 3 to both sides, which simplifies to −2x≤12-2x \leq 12. Next, divide both sides by -2. A critical rule of inequalities is that dividing or multiplying by a negative number requires reversing the inequality symbol. Therefore, ≤\leq becomes ≥\geq, and 12÷−2=−612 \div -2 = -6, resulting in the final answer of x≥−6x \geq -6.

Step-by-Step Solution

1
Isolate the variable term by adding 3 to both sides of the inequality.
−2x≤12-2x \leq 12
This removes the constant term from the side with the variable, leaving only the term with x.
2
Divide both sides by -2 and reverse the inequality symbol.
x≥−6x \geq -6
When multiplying or dividing an inequality by a negative number, the direction of the inequality must flip to keep the mathematical statement true.

Key Concept

Solving one-variable inequalities and applying the inequality flip rule.
Estimated Time:45s
Question 14Question →

A student needs to solve the following algebraic inequality:

7−2(x−3)≥4x−57 - 2(x - 3) \geq 4x - 5

What is the correct solution for xx?

Show answer & explanation

Answer: x≤3x \leq 3

Answer

The correct solution is x≤3x \leq 3.
The correct answer is derived by carefully distributing the −2-2, combining the constants, grouping the variables on one side, and dividing by −6-6. Crucially, dividing by a negative coefficient triggers the rule to flip the ≥\geq sign to ≤\leq, resulting in x≤3x \leq 3.

Step-by-Step Solution

1
Distribute the −2-2 into the parentheses on the left side of the inequality.
7−2x+6≥4x−57 - 2x + 6 \geq 4x - 5
To eliminate the parentheses and allow like terms to be combined.
2
Combine the constant terms on the left side.
13−2x≥4x−513 - 2x \geq 4x - 5
Adding 77 and 66 simplifies the left side of the algebraic expression.
3
Subtract 4x4x from both sides of the inequality.
13−6x≥−513 - 6x \geq -5
To group all terms containing the variable xx onto the left side.
4
Subtract 1313 from both sides.
−6x≥−18-6x \geq -18
To isolate the variable term on one side and the constant values on the other.
5
Divide both sides by −6-6 and reverse the inequality symbol.
x≤3x \leq 3
Whenever an inequality is multiplied or divided by a negative number, the direction of the inequality sign must be flipped.

Key Concept

Solving multi-step inequalities, including the distributive property and the inequality flip rule for negative coefficients.
Question 15Question →

The value x=−2x = -2 is a valid solution to the inequality −4(x−3)≤2x+18-4(x - 3) \leq 2x + 18.

Show answer & explanation

Answer: False

Answer

False
The statement is false because substituting −2-2 into the inequality results in 20≤1420 \leq 14, which is not mathematically accurate. Alternatively, algebraic solving yields x≥−1x \geq -1, and −2-2 does not satisfy this condition.

Step-by-Step Solution

1
Substitute x=−2x = -2 into the left side of the inequality.
−4(−2−3)=−4(−5)=20-4(-2 - 3) = -4(-5) = 20
To determine the numerical value of the left expression when x=−2x = -2.
2
Substitute x=−2x = -2 into the right side of the inequality.
2(−2)+18=−4+18=142(-2) + 18 = -4 + 18 = 14
To determine the numerical value of the right expression when x=−2x = -2.
3
Compare the two results using the given inequality symbol.
The statement 20≤1420 \leq 14 is false.
To verify if the substituted value satisfies the mathematical condition.

Key Concept

Solving One-Variable Inequalities
Question 16Question →
7(y−2)−4y>5y+107(y - 2) - 4y > 5y + 10

What are all the possible values of yy that satisfy the inequality shown above?

Show answer & explanation

Answer: y<−12y < -12

Answer

The correct solution set is y<−12y < -12.
The correct answer is found by properly applying the distributive property, grouping like terms, isolating the variable, and correctly reversing the inequality sign when dividing by -2.

Step-by-Step Solution

1
Distribute the 7 to both terms inside the parentheses.
7y−14−4y>5y+107y - 14 - 4y > 5y + 10
This clears the parentheses, allowing the variable terms to be grouped.
2
Combine the like terms on the left side of the inequality.
3y−14>5y+103y - 14 > 5y + 10
Subtracting 4y4y from 7y7y simplifies the left side to 3y3y.
3
Subtract 5y5y from both sides to group the variable terms on the left.
−2y−14>10-2y - 14 > 10
This moves all terms containing yy to one side of the inequality.
4
Add 14 to both sides to isolate the variable term.
−2y>24-2y > 24
This leaves the −2y-2y term by itself on the left side.
5
Divide both sides by -2 and reverse the inequality sign.
y<−12y < -12
According to the properties of inequalities, multiplying or dividing by a negative number requires flipping the direction of the inequality symbol.

Key Concept

Solving One-Variable Inequalities
Question 17Question →

Evaluate the following mathematical statement:

−4x+10≥−14-4x + 10 \geq -14

Which of the following represents the correct solution for xx?

Show answer & explanation

Answer: x≤6x \leq 6

Answer

The correct answer is x≤6x \leq 6.
To solve the inequality, first subtract 1010 from both sides to get −4x≥−24-4x \geq -24. Next, isolate xx by dividing both sides by −4-4. Because you are dividing by a negative number, you must flip the inequality sign from ≥\geq to ≤\leq, resulting in x≤6x \leq 6.

Step-by-Step Solution

1
Subtract 1010 from both sides of the inequality.
−4x≥−24-4x \geq -24
This isolates the term containing the variable xx on one side of the inequality.
2
Divide both sides by −4-4 and reverse the inequality sign.
x≤6x \leq 6
When multiplying or dividing an inequality by a negative number, the direction of the inequality symbol must be flipped to maintain a true statement.

Key Concept

Solving one-variable inequalities requires the same algebraic steps as solving linear equations, with the critical exception that the inequality sign must be reversed when multiplying or dividing by a negative number.
Question 18Question →

What is the greatest integer value of xx that satisfies the following inequality?

3(4−2x)−2x>5(x−3)−10x3(4 - 2x) - 2x > 5(x - 3) - 10x
Show answer & explanation

Answer: 8

Answer

8
Solving the inequality yields x < 9. Since the inequality is strictly less than 9, the greatest integer that satisfies the condition is 8.

Step-by-Step Solution

1
Distribute the 3 and 5
12−6x−2x>5x−15−10x12 - 6x - 2x > 5x - 15 - 10x
To eliminate the parentheses on both sides of the inequality.
2
Combine like terms
12−8x>−5x−1512 - 8x > -5x - 15
To simplify both sides before isolating the variable.
3
Isolate the variable terms
−3x>−27-3x > -27
Subtract 12 from both sides and add 5x to both sides to get all x terms on the left and constants on the right.
4
Divide by -3
x<9x < 9
Dividing an inequality by a negative number requires flipping the inequality sign.
5
Find the greatest integer
8
The inequality states x must be strictly less than 9, so the largest possible integer is 8.

Key Concept

Solving linear inequalities, applying the negative division flip rule, and interpreting integer boundaries.
Question 19Question →
What is the minimum integer value of xx that satisfies the following inequality?
−3x+2≤14-3x + 2 \leq 14
Show answer & explanation

Answer: -4

Answer

-4
By isolating the variable and correctly reversing the inequality sign when dividing by a negative number, we find the solution set is x≥−4x \geq -4. The smallest integer in this set is -4.

Step-by-Step Solution

1
Subtract 2 from both sides of the inequality.
−3x≤12-3x \leq 12
To isolate the variable term.
2
Divide both sides by -3 and reverse the inequality sign.
x≥−4x \geq -4
Dividing an inequality by a negative number requires reversing the direction of the inequality symbol to maintain the true mathematical relationship.
3
Find the minimum integer value that satisfies the solution set.
-4
The solution set includes all numbers greater than or equal to -4. The smallest integer in this set is -4.

Key Concept

Solving two-step linear inequalities and applying the rule for dividing by a negative number.
Question 20Question →

Solve the following inequality for xx:

−2x+6≤14-2x + 6 \leq 14

Which of the following is the correct solution?

Show answer & explanation

Answer: x≥−4x \geq -4

Answer

The correct solution is x≥−4x \geq -4.
The correct answer is found by subtracting 6 from 14 to get 8, and then dividing 8 by -2. Because we divided by a negative number, the less than or equal to symbol (≤\leq) must flip to a greater than or equal to symbol (≥\geq), yielding x≥−4x \geq -4.

Step-by-Step Solution

1
Subtract 6 from both sides of the inequality to isolate the variable term.
−2x≤14−6-2x \leq 14 - 6
−2x≤8-2x \leq 8
This is the first step in using inverse operations to isolate xx.
2
Divide both sides by −2-2 and reverse the inequality symbol.
x≥8÷(−2)x \geq 8 \div (-2)
x≥−4x \geq -4
When multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be flipped.

Key Concept

Solving One-Variable Inequalities
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