Examkin

Elementary Algebra

302 questions

All types302Multiple choice155Numeric125Matching15Fill in the blank7
Question 261Question →

A laboratory chamber's initial temperature is 72\frac{7}{2} degrees Celsius, and it decreases at a constant rate of 54\frac{5}{4} degrees Celsius per hour. The temperature of the chamber must reach at most −12-\frac{1}{2} degrees Celsius to complete an experiment. Which of the following inequalities represents the number of hours, hh, the experiment must run to reach this temperature?

Show answer & explanation

Answer: h≥165h \geq \frac{16}{5}

Answer

h≥165h \geq \frac{16}{5}
The correct answer is the inequality stating that hh must be greater than or equal to sixteen-fifths. This is found by setting up the linear inequality 72−54h≤−12\frac{7}{2} - \frac{5}{4}h \leq -\frac{1}{2}, subtracting 72\frac{7}{2} from both sides to get −54h≤−4-\frac{5}{4}h \leq -4, and then multiplying by −45-\frac{4}{5} while reversing the inequality sign.

Step-by-Step Solution

1
Set up the inequality representing the temperature constraint.
72−54h≤−12\frac{7}{2} - \frac{5}{4}h \leq -\frac{1}{2}
The initial temperature is 72\frac{7}{2}, the rate of decrease is 54\frac{5}{4} per hour hh, and the final temperature must be at most −12-\frac{1}{2}.
2
Subtract 72\frac{7}{2} from both sides of the inequality.
−54h≤−4-\frac{5}{4}h \leq -4
This isolates the variable term on the left side of the inequality. The subtraction is −12−72=−82=−4-\frac{1}{2} - \frac{7}{2} = -\frac{8}{2} = -4.
3
Multiply both sides by −45-\frac{4}{5} and reverse the inequality sign.
h≥165h \geq \frac{16}{5}
Multiplying or dividing by a negative number requires reversing the direction of the inequality sign. The calculation is −4×(−45)=165-4 \times \left(-\frac{4}{5}\right) = \frac{16}{5}.

Key Concept

Solving linear inequalities involving negative coefficients and applying the inequality sign-flip rule.
Estimated Time:1m 30s
Question 262Question →

For all real values of xx that satisfy the inequality 5−3x−24≥x+625 - \frac{3x - 2}{4} \ge \frac{x + 6}{2}, the solution set is represented by x≤bx \le b. What is the value of bb?

Show answer & explanation

Answer: 2

Answer

The value of bb is 2.
To solve 5−3x−24≥x+625 - \frac{3x - 2}{4} \ge \frac{x + 6}{2}, multiply all terms by 4 to clear the denominators, resulting in 20−(3x−2)≥2(x+6)20 - (3x - 2) \ge 2(x + 6). Carefully distribute the negative sign to obtain 20−3x+2≥2x+1220 - 3x + 2 \ge 2x + 12. Simplifying the left side gives 22−3x≥2x+1222 - 3x \ge 2x + 12. Moving the variable terms to one side yields 10≥5x10 \ge 5x, which simplifies to x≤2x \le 2. Thus, the upper bound value bb is 2.

Step-by-Step Solution

1
Multiply both sides of the inequality by the least common denominator, which is 4.
20−(3x−2)≥2(x+6)20 - (3x - 2) \ge 2(x + 6)
Multiplying all terms by the common denominator eliminates fractions and simplifies the inequality.
2
Distribute the negative sign to the numerator terms on the left and distribute the 2 on the right.
20−3x+2≥2x+1220 - 3x + 2 \ge 2x + 12
Distributing the negative sign across (3x−2)(3x - 2) changes it to −3x+2-3x + 2. Distributing 2 across (x+6)(x + 6) yields 2x+122x + 12.
3
Combine like terms on the left side of the inequality.
22−3x≥2x+1222 - 3x \ge 2x + 12
Combining the constant terms 2020 and 22 simplifies the expression to 2222.
4
Isolate the variable terms by adding 3x3x and subtracting 12 from both sides.
10≥5x10 \ge 5x
Grouping variables on one side and constants on the other allows us to solve for xx.
5
Divide both sides by 5.
2≥x2 \ge x (or x≤2x \le 2)
Dividing by a positive number isolates the variable without changing the direction of the inequality sign.

Key Concept

Solving linear inequalities involving fractions and distributing negative coefficients.

Alternative Method

We can write the inequality by separating each fraction term first: 5−34x+24≥12x+625 - \frac{3}{4}x + \frac{2}{4} \ge \frac{1}{2}x + \frac{6}{2}. This simplifies to 5.5−0.75x≥0.5x+35.5 - 0.75x \ge 0.5x + 3. Subtracting 0.5x0.5x and 5.55.5 from both sides gives −1.25x≥−2.5-1.25x \ge -2.5. Dividing by −1.25-1.25 and reversing the inequality sign gives x≤2x \le 2.
Estimated Time:1m 30s
Question 263Question →

A homeowner has a square patio with a side length of 3x−23x - 2 feet. She decides to cover a portion of the patio with a rectangular outdoor rug that has dimensions 2x−12x - 1 feet by x+4x + 4 feet. Which of the following expressions represents the area, in square feet, of the patio that remains uncovered by the rug?

Show answer & explanation

Answer: 7x2−19x+87x^2 - 19x + 8

Answer

7x2−19x+87x^2 - 19x + 8
The expression representing the uncovered area is obtained by subtracting the area of the rug from the area of the patio. First, the area of the square patio is calculated as (3x−2)2=9x2−12x+4(3x - 2)^2 = 9x^2 - 12x + 4. Second, the area of the rug is calculated as (2x−1)(x+4)=2x2+7x−4(2x - 1)(x + 4) = 2x^2 + 7x - 4. Subtracting the rug's area from the patio's area requires distributing the negative sign: (9x2−12x+4)−(2x2+7x−4)=9x2−12x+4−2x2−7x+4(9x^2 - 12x + 4) - (2x^2 + 7x - 4) = 9x^2 - 12x + 4 - 2x^2 - 7x + 4. Combining like terms yields 7x2−19x+87x^2 - 19x + 8.

Step-by-Step Solution

1
Calculate the area of the square patio by squaring its side length.
Apatio=(3x−2)2=9x2−12x+4A_{\text{patio}} = (3x - 2)^2 = 9x^2 - 12x + 4
The area of a square is equal to the square of its side length: Area=s2\text{Area} = s^2.
2
Calculate the area of the rectangular rug by multiplying its length and width.
Arug=(2x−1)(x+4)=2x2+8x−x−4=2x2+7x−4A_{\text{rug}} = (2x - 1)(x + 4) = 2x^2 + 8x - x - 4 = 2x^2 + 7x - 4
The area of a rectangle is equal to the product of its length and width: Area=l×w\text{Area} = l \times w.
3
Subtract the area of the rug from the area of the patio, distributing the negative sign to all terms of the rug's area.
Auncovered=(9x2−12x+4)−(2x2+7x−4)=9x2−12x+4−2x2−7x+4=7x2−19x+8A_{\text{uncovered}} = (9x^2 - 12x + 4) - (2x^2 + 7x - 4) = 9x^2 - 12x + 4 - 2x^2 - 7x + 4 = 7x^2 - 19x + 8
Subtracting a polynomial requires distributing the negative sign to every term inside the parentheses and then combining like terms.

Key Concept

Subtracting one polynomial from another requires distributing the negative sign to every term of the subtracted polynomial before combining like terms.
Estimated Time:1m 30s
Question 264Question →

A chemical solution in a laboratory has an initial volume of 5050 milliliters and evaporates at a rate of 72\frac{7}{2} milliliters per hour. A second chemical solution has an initial volume of 2020 milliliters and evaporates at a rate of 54\frac{5}{4} milliliters per hour. After how many hours, tt, will the volume of the first solution be at most the volume of the second solution?

Show answer & explanation

Answer: t≥403t \ge \frac{40}{3}

Answer

The correct solution is the set of values t≥403t \ge \frac{40}{3}.
Translating the problem statement gives the inequality 50−72t≤20−54t50 - \frac{7}{2}t \le 20 - \frac{5}{4}t. Subtracting 50 from both sides and adding 54t\frac{5}{4}t to both sides results in −94t≤−30-\frac{9}{4}t \le -30. Multiplying both sides by −49-\frac{4}{9} and reversing the inequality sign because of the negative multiplier yields the correct solution, t≥403t \ge \frac{40}{3}.

Step-by-Step Solution

1
Write the inequality representing the physical situation.
50−72t≤20−54t50 - \frac{7}{2}t \le 20 - \frac{5}{4}t
The volume of the first solution after tt hours is 50−72t50 - \frac{7}{2}t, and the volume of the second solution is 20−54t20 - \frac{5}{4}t. The phrase 'at most' means the first volume must be less than or equal to the second volume.
2
Isolate the variable terms on the left side and the constant terms on the right side.
−72t+54t≤20−50-\frac{7}{2}t + \frac{5}{4}t \le 20 - 50
Grouping like terms makes it possible to simplify both sides of the inequality.
3
Find a common denominator of 4 to combine the fractions on the left side, and simplify the constant terms on the right side.
−94t≤−30-\frac{9}{4}t \le -30
Converting −72t-\frac{7}{2}t to −144t-\frac{14}{4}t allows us to add it to 54t\frac{5}{4}t, resulting in −94t-\frac{9}{4}t.
4
Multiply both sides of the inequality by −49-\frac{4}{9} to solve for tt, reversing the inequality sign because we are multiplying by a negative number.
t≥403t \ge \frac{40}{3}
Multiplying by the reciprocal of the coefficient isolates tt. The inequality sign must be reversed (≤{\le} to ≥{\ge}) because we are multiplying by a negative value.

Key Concept

Solving Linear Inequalities
Estimated Time:1m 30s
Question 265Question →

A landscape architect designs a square courtyard with a side length of 4x−34x - 3 meters. A square garden bed with a side length of 2x−12x - 1 meters is placed in one of the corners of the courtyard. The remaining area of the courtyard is paved. If the paved area, in square meters, is represented by the polynomial ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants, what is the value of bb?

Show answer & explanation

Answer: -20

Answer

The value of bb is −20-20.
The remaining paved area is the difference between the area of the courtyard and the garden bed: (16x2−24x+9)−(4x2−4x+1)=12x2−20x+8(16x^2 - 24x + 9) - (4x^2 - 4x + 1) = 12x^2 - 20x + 8. The coefficient of the linear xx term is −20-20.

Step-by-Step Solution

1
Find the polynomial representing the total area of the courtyard.
Atotal=(4x−3)2=16x2−24x+9A_{\text{total}} = (4x - 3)^2 = 16x^2 - 24x + 9 square meters
The area of a square is the square of its side length. We expand (4x−3)2(4x - 3)^2 using the identity (p−q)2=p2−2pq+q2(p - q)^2 = p^2 - 2pq + q^2.
2
Find the polynomial representing the area of the garden bed.
Agarden=(2x−1)2=4x2−4x+1A_{\text{garden}} = (2x - 1)^2 = 4x^2 - 4x + 1 square meters
The garden bed is also a square, so we expand (2x−1)2(2x - 1)^2 using the binomial squaring identity.
3
Subtract the garden bed area from the total area to find the paved area.
Apaved=12x2−20x+8A_{\text{paved}} = 12x^2 - 20x + 8 square meters
The paved area is the difference between the two areas. We distribute the negative sign to all terms of the subtracted polynomial: −(4x2−4x+1)=−4x2+4x−1-(4x^2 - 4x + 1) = -4x^2 + 4x - 1, and then combine like terms.
4
Identify the coefficient of the xx term, bb.
b=−20b = -20
Comparing 12x2−20x+812x^2 - 20x + 8 to ax2+bx+cax^2 + bx + c, the coefficient of the linear xx term is −20-20.

Key Concept

Operations on Polynomials

Alternative Method

To find only the coefficient of the linear term, expand and subtract only the linear terms from both binomial expansions: 2(4x)(−3)−2(2x)(−1)=−24x−(−4x)=−20x2(4x)(-3) - 2(2x)(-1) = -24x - (-4x) = -20x. The coefficient bb is therefore −20-20.
Estimated Time:1m 30s
Question 266Question →

A square metal sheet has a side length of 3x33x^3 inches. A square cutout with a side length of (x3−2)(x^3 - 2) inches is removed from the center of the sheet. Which of the following polynomial expressions represents the area, in square inches, of the remaining metal sheet?

Show answer & explanation

Answer: 8x6+4x3−48x^6 + 4x^3 - 4

Answer

The expression 8x6+4x3−48x^6 + 4x^3 - 4 represents the remaining area of the metal sheet.
To find the remaining area, subtract the area of the square cutout from the total area of the metal sheet. The total area is the square of the side length 3x33x^3, which is (3x3)2=9x6(3x^3)^2 = 9x^6. The area of the cutout is the square of (x3−2)(x^3 - 2), which is (x3−2)2=x6−4x3+4(x^3 - 2)^2 = x^6 - 4x^3 + 4. Subtracting the cutout area from the total area yields 9x6−(x6−4x3+4)=9x6−x6+4x3−4=8x6+4x3−49x^6 - (x^6 - 4x^3 + 4) = 9x^6 - x^6 + 4x^3 - 4 = 8x^6 + 4x^3 - 4. This matches the correct expression.

Step-by-Step Solution

1
Calculate the total area of the square metal sheet.
Areatotal=(3x3)2=9x6Area_{total} = (3x^3)^2 = 9x^6
The area of a square is the square of its side length. According to the power of a product rule, (ab)n=anbn(ab)^n = a^n b^n, and the power of a power rule, (xa)b=xab(x^a)^b = x^{ab}.
2
Calculate the area of the square cutout.
Areacutout=(x3−2)2=x6−4x3+4Area_{cutout} = (x^3 - 2)^2 = x^6 - 4x^3 + 4
Squaring the binomial yields (x3)2−2(2)(x3)+22(x^3)^2 - 2(2)(x^3) + 2^2, which simplifies to x6−4x3+4x^6 - 4x^3 + 4.
3
Subtract the area of the cutout from the total area of the metal sheet.
Arearemaining=9x6−(x6−4x3+4)=8x6+4x3−4Area_{remaining} = 9x^6 - (x^6 - 4x^3 + 4) = 8x^6 + 4x^3 - 4
Distributing the negative sign through the parentheses yields 9x6−x6+4x3−49x^6 - x^6 + 4x^3 - 4. Combining the like terms 9x69x^6 and −x6-x^6 gives the final simplified expression.

Key Concept

Polynomial subtraction and binomial expansion
Estimated Time:1m 30s
Question 267Question →

What is the maximum value of xx that satisfies the inequality 2−5x3≥2x+8\frac{2 - 5x}{3} \ge 2x + 8?

Show answer & explanation

Answer: -2

Answer

-2
Solving the inequality by isolating xx leads to x≤−2x \le -2, meaning that any value of xx less than or equal to −2-2 satisfies the inequality. Therefore, the maximum possible value is −2-2.

Step-by-Step Solution

1
Multiply both sides of the inequality by 3 to clear the denominator.
2−5x≥6x+242 - 5x \ge 6x + 24
Multiplying by a positive number eliminates the fraction without changing the inequality direction.
2
Subtract 6x6x from both sides of the inequality.
2−11x≥242 - 11x \ge 24
This gathers the variable terms on the left side of the inequality.
3
Subtract 2 from both sides of the inequality.
−11x≥22-11x \ge 22
This isolates the variable term on the left side.
4
Divide both sides by −11-11 and reverse the inequality sign.
x≤−2x \le -2
Dividing by a negative number requires reversing the direction of the inequality sign. The resulting inequality defines the upper bound for xx.

Key Concept

Solving linear inequalities and applying the sign-flip rule when multiplying or dividing by a negative number.
Question 268Question →

For all real values of xx, which of the following inequalities represents the complete solution set to 7−3(x−4)≥5x+317 - 3(x - 4) \ge 5x + 31?

Show answer & explanation

Answer: x≤−32x \le -\frac{3}{2}

Answer

The inequality that represents the complete solution set is x≤−32x \le -\frac{3}{2}
The correct answer is the inequality stating that xx is less than or equal to negative three-halves. Simplifying the expression yields −8x≥12-8x \ge 12, and dividing by −8-8 correctly reverses the inequality direction to produce x≤−32x \le -\frac{3}{2}.

Step-by-Step Solution

1
Distribute −3-3 to both terms inside the parentheses on the left side of the inequality.
7−3x+12≥5x+317 - 3x + 12 \ge 5x + 31
To eliminate the parentheses and prepare the inequality for simplification.
2
Combine the constant terms 77 and 1212 on the left side.
19−3x≥5x+3119 - 3x \ge 5x + 31
To group like terms on the left side.
3
Subtract 5x5x from both sides of the inequality to collect variable terms on one side.
19−8x≥3119 - 8x \ge 31
To isolate the variable term on the left side.
4
Subtract 1919 from both sides of the inequality.
−8x≥12-8x \ge 12
To isolate the variable term completely before division.
5
Divide both sides of the inequality by −8-8 and flip the inequality sign.
x≤−128x \le -\frac{12}{8}, which simplifies to x≤−32x \le -\frac{3}{2}
Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.

Key Concept

Solving multi-step linear inequalities, specifically reversing the inequality sign when multiplying or dividing by a negative number.
Question 269Question →

For all real values of nn that satisfy the inequality 3−2n4<3n−225\frac{3 - 2n}{4} < \frac{3n - 22}{5}, what is the smallest possible integer value of nn?

Show answer & explanation

Answer: 5

Answer

The smallest integer value of nn that satisfies the inequality is 5.
Solving the inequality yields n>10322≈4.68n > \frac{103}{22} \approx 4.68. The smallest integer greater than 4.684.68 is 55.

Step-by-Step Solution

1
Multiply both sides by 20 to clear the denominators.
5(3−2n)<4(3n−22)5(3 - 2n) < 4(3n - 22)
To eliminate the fractions and simplify the inequality.
2
Distribute the constants on both sides.
15−10n<12n−8815 - 10n < 12n - 88
To remove the parentheses.
3
Subtract 12n12n from both sides.
15−22n<−8815 - 22n < -88
To group the terms containing the variable on the left side.
4
Subtract 15 from both sides.
−22n<−103-22n < -103
To isolate the variable term on the left side.
5
Divide both sides by −22-22 and reverse the inequality sign.
n>10322n > \frac{103}{22}
Dividing by a negative number reverses the direction of the inequality sign.
6
Convert the fraction to a decimal to identify the boundary.
n>4.68n > 4.68
To find the smallest integer value that satisfies this condition.
7
Identify the smallest integer greater than 4.684.68.
55
The smallest integer greater than 4.684.68 is 55.

Key Concept

Solving linear inequalities by applying inverse operations and reversing the inequality sign when multiplying or dividing by a negative number.
Estimated Time:1m 30s
Question 270Question →

Which of the following inequalities represents the complete solution set for xx in the inequality x3−2x−52≥2\frac{x}{3} - \frac{2x - 5}{2} \ge 2?

Show answer & explanation

Answer: x≤34x \le \frac{3}{4}

Answer

The inequality is satisfied by all values of xx less than or equal to 34\frac{3}{4}.
The correct solution is obtained by rewriting the fractions with a common denominator of 6, which yields 2x−3(2x−5)6≥2\frac{2x - 3(2x - 5)}{6} \ge 2. Distributing the negative sign gives −4x+156≥2\frac{-4x + 15}{6} \ge 2. Multiplying both sides by 6 and subtracting 15 results in −4x≥−3-4x \ge -3. Dividing by −4-4 and reversing the inequality sign yields x≤34x \le \frac{3}{4}.

Step-by-Step Solution

1
Find a common denominator of 6 for the two fractions on the left side of the inequality.
2x6−3(2x−5)6≥2\frac{2x}{6} - \frac{3(2x - 5)}{6} \ge 2
Finding a common denominator is necessary to combine fractional terms with different denominators.
2
Combine the fractions into a single expression, distributing the subtraction sign to both terms in the numerator of the second fraction.
2x−(6x−15)6≥2  ⟹  −4x+156≥2\frac{2x - (6x - 15)}{6} \ge 2 \implies \frac{-4x + 15}{6} \ge 2
Combining the fractions simplifies the inequality. Distributing the negative sign ensures correct operations on the terms.
3
Multiply both sides of the inequality by 6.
−4x+15≥12-4x + 15 \ge 12
Multiplying by a positive number isolates the numerator without changing the direction of the inequality sign.
4
Subtract 15 from both sides of the inequality.
−4x≥−3-4x \ge -3
This isolates the variable term on the left side.
5
Divide both sides by −4-4 and reverse the direction of the inequality sign.
x≤34x \le \frac{3}{4}
Dividing both sides of an inequality by a negative number requires reversing the inequality sign to maintain the correct relationship.

Key Concept

Solving linear inequalities requires applying standard algebraic operations while remembering to reverse the inequality sign when multiplying or dividing both sides by a negative number.

Alternative Method

Instead of combining fractions first, multiply every term on both sides of the inequality by 6 (the least common multiple of the denominators) to eliminate the fractions immediately: 6⋅(x3)−6⋅(2x−52)≥6⋅26 \cdot \left(\frac{x}{3}\right) - 6 \cdot \left(\frac{2x - 5}{2}\right) \ge 6 \cdot 2. This simplifies directly to 2x−3(2x−5)≥122x - 3(2x - 5) \ge 12, which reduces to −4x+15≥12-4x + 15 \ge 12.
Estimated Time:1m 30s
Question 271Question →

A warehouse is cooling a refrigerated storage room. The initial temperature of the room is 22∘C22^\circ\text{C}. A cooling system lowers the temperature by 1.8∘C1.8^\circ\text{C} per hour. To store a specific vaccine, the temperature of the room must be kept strictly below −5∘C-5^\circ\text{C}. What is the minimum number of whole hours the cooling system must run to reach a safe storage temperature?

Show answer & explanation

Answer: 16

Answer

The minimum number of whole hours the cooling system must run is 16.
To find the minimum number of whole hours, we set up the inequality 22−1.8h<−522 - 1.8h < -5. Subtracting 22 from both sides yields −1.8h<−27-1.8h < -27. Dividing by the negative rate −1.8-1.8 requires reversing the inequality sign, which gives h>15h > 15. The smallest integer greater than 15 is 16.

Step-by-Step Solution

1
Set up the inequality representing the temperature condition.
22−1.8h<−522 - 1.8h < -5
The initial temperature is 22∘C22^\circ\text{C}, and it decreases by 1.8∘C1.8^\circ\text{C} per hour hh. The temperature must be strictly below −5∘C-5^\circ\text{C}.
2
Isolate the variable term by subtracting 22 from both sides.
−1.8h<−27-1.8h < -27
Subtracting 22 simplifies the left side and groups the constant terms on the right side.
3
Divide both sides by -1.8 and reverse the inequality sign.
h>15h > 15
Dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.
4
Determine the smallest integer value for h that satisfies the inequality.
16
Since the time must be strictly greater than 15 hours, the smallest whole number of hours that satisfies this is 16.

Key Concept

Solving linear inequalities by isolating the variable and reversing the inequality sign when dividing by a negative number.
Question 272Question →

What is the complete set of real values of xx that satisfy the inequality 34−2x3>2\frac{3}{4} - \frac{2x}{3} > 2?

Show answer & explanation

Answer: x<−158x < -\frac{15}{8}

Answer

x<−158x < -\frac{15}{8}
Subtracting 34\frac{3}{4} from both sides of the inequality gives −2x3>54-\frac{2x}{3} > \frac{5}{4}. Multiplying both sides by −32-\frac{3}{2} and reversing the inequality sign from greater than (>>) to less than (<<) yields x<−158x < -\frac{15}{8}.

Step-by-Step Solution

1
Subtract 34\frac{3}{4} from both sides of the inequality.
−2x3>54-\frac{2x}{3} > \frac{5}{4}
To isolate the term containing the variable xx on one side of the inequality.
2
Multiply both sides of the inequality by −32-\frac{3}{2} and reverse the inequality sign.
x<−158x < -\frac{15}{8}
Multiplying or dividing an inequality by a negative number requires reversing the direction of the inequality sign.

Key Concept

Solving linear inequalities by isolating the variable and applying the sign reversal rule when multiplying or dividing by a negative value.
Question 273Question →

What is the greatest integer value of xx that satisfies the inequality 7−5x3−4≥2(x+6)\frac{7 - 5x}{3} - 4 \ge 2(x + 6)?

Show answer & explanation

Answer: -4

Answer

The greatest integer value of xx that satisfies the inequality is −4-4.
To find the greatest integer value of xx that satisfies the inequality, solve the inequality algebraically. First, clear the fraction by multiplying all terms by 3: 7−5x−12≥6(x+6)7 - 5x - 12 \ge 6(x + 6). Simplify the left side to −5x−5-5x - 5 and distribute the right side to get 6x+366x + 36. Move variables to the left side by subtracting 6x6x to get −11x−5≥36-11x - 5 \ge 36. Add 5 to both sides to get −11x≥41-11x \ge 41. Finally, divide by −11-11 and reverse the inequality sign, yielding x≤−4111x \le -\frac{41}{11}, which is approximately x≤−3.73x \le -3.73. The greatest integer less than or equal to −3.73-3.73 is −4-4.

Step-by-Step Solution

1
Multiply the entire inequality by 3 to clear the fraction
7−5x−12≥6(x+6)7 - 5x - 12 \ge 6(x + 6)
Multiplying by the common denominator eliminates the fraction and simplifies further algebraic steps.
2
Simplify the left side and distribute the right side
−5x−5≥6x+36-5x - 5 \ge 6x + 36
Combines constant terms on the left side and expands the parentheses on the right side.
3
Subtract 6x6x from both sides
−11x−5≥36-11x - 5 \ge 36
Gathers all variable terms on the left side of the inequality.
4
Add 5 to both sides
−11x≥41-11x \ge 41
Isolates the variable term by moving the constant to the right side.
5
Divide by -11 and reverse the inequality sign
x≤−4111x \le -\frac{41}{11}
Isolates the variable xx. Reversing the inequality sign is required when multiplying or dividing both sides by a negative number.
6
Determine the greatest integer satisfying the inequality
x≤−3.73x \le -3.73, so the greatest integer is −4-4
Since −4111≈−3.73-\frac{41}{11} \approx -3.73, the largest integer that is less than or equal to this value is −4-4.

Key Concept

Solving linear inequalities, applying the sign-flip rule when dividing by a negative number, and identifying integer boundaries.
Question 274Question →

Evaluate the algebraic expression for the given variable values.

Fill in the blanks below

If x=−3x = -3 and y=4y = 4, the value of the expression 2x2−3y2x^2 - 3y is .
Show answer & explanation

Answer

6
Substituting x=−3x = -3 into x2x^2 gives (−3)2=9(-3)^2 = 9. Multiplying by 22 yields 1818. Substituting y=4y = 4 into 3y3y gives 1212. Subtracting 1212 from 1818 gives the correct value of 66.

Step-by-Step Solution

1
Substitute the given values x=−3x = -3 and y=4y = 4 into the expression 2x2−3y2x^2 - 3y.
2(−3)2−3(4)2(-3)^2 - 3(4)
Replace each variable with its designated numeric value.
2
Evaluate the exponent (−3)2(-3)^2.
2(9)−3(4)2(9) - 3(4)
Following the order of operations (PEMDAS), exponents are evaluated before multiplication.
3
Perform the multiplication operations.
18−1218 - 12
Multiply 2×9=182 \times 9 = 18 and 3×4=123 \times 4 = 12.
4
Subtract the terms to find the final value.
66
Perform final subtraction: 18−12=618 - 12 = 6.

Key Concept

Evaluating Algebraic Expressions with Negative Values
Estimated Time:45s
Question 275Question →

If a=−4a = -4 and b=3b = 3, what is the value of the expression 3a2−2ab3a^2 - 2ab?

Show answer & explanation

Answer: 72

Answer

72
Substituting a=−4a = -4 and b=3b = 3 yields 3(−4)2−2(−4)(3)=3(16)−(−24)=48+24=723(-4)^2 - 2(-4)(3) = 3(16) - (-24) = 48 + 24 = 72.

Step-by-Step Solution

1
Substitute the given values into the expression
Replace aa with −4-4 and bb with 33 in 3a2−2ab3a^2 - 2ab to get 3(−4)2−2(−4)(3)3(-4)^2 - 2(-4)(3).
Direct substitution of known variable values.
2
Evaluate the exponent
(−4)2=16(-4)^2 = 16, so the term 3(−4)23(-4)^2 becomes 3(16)=483(16) = 48.
Exponents must be evaluated before multiplication according to PEMDAS.
3
Evaluate the second multiplication term
2ab=2(−4)(3)=−242ab = 2(-4)(3) = -24.
Multiply the numerical factors together.
4
Subtract the terms to find the final value
48−(−24)=48+24=7248 - (-24) = 48 + 24 = 72.
Subtracting a negative number is equivalent to adding its positive value.

Key Concept

Evaluating Algebraic Expressions with Negative Values
Estimated Time:45s
Question 276Question →

If a=−4a = -4 and b=7b = 7, what is the value of the algebraic expression (a+b)2−3a(a + b)^2 - 3a?

Show answer & explanation

Answer: 21

Answer

21
Substituting a=−4a = -4 and b=7b = 7 into (a+b)2−3a(a + b)^2 - 3a gives ((−4)+7)2−3(−4)=(3)2+12=9+12=21((-4) + 7)^2 - 3(-4) = (3)^2 + 12 = 9 + 12 = 21.

Step-by-Step Solution

1
Substitute the given values into the expression
((−4)+7)2−3(−4)((-4) + 7)^2 - 3(-4)
Replace aa with −4-4 and bb with 77.
2
Simplify inside the parentheses
(3)2−3(−4)(3)^2 - 3(-4)
Following order of operations (PEMDAS), simplify −4+7=3-4 + 7 = 3 first.
3
Evaluate the exponent and multiplication
9+129 + 12
Square 33 to get 99, and multiply −3-3 by −4-4 to get +12+12.
4
Add the terms together
21
Combine 99 and 1212.

Key Concept

Evaluating Algebraic Expressions
Question 277Question →

If x=−2x = -2 and y=5y = 5, what is the value of the algebraic expression x3+4yx+y\frac{x^3 + 4y}{x + y}?

Show answer & explanation

Answer: 4

Answer

The value of the expression is 4.
Substituting x=−2x = -2 and y=5y = 5 yields (−2)3+4(5)=−8+20=12(-2)^3 + 4(5) = -8 + 20 = 12 in the numerator, and −2+5=3-2 + 5 = 3 in the denominator. Dividing 1212 by 33 gives 44.

Step-by-Step Solution

1
Substitute the given numerical values into the algebraic expression.
(−2)3+4(5)(−2)+5\frac{(-2)^3 + 4(5)}{(-2) + 5}
Replace each occurrence of xx with −2-2 and yy with 55.
2
Evaluate the numerator using order of operations.
−8+20=12-8 + 20 = 12
Calculate (−2)3=−8(-2)^3 = -8 and 4(5)=204(5) = 20, then add the terms together.
3
Evaluate the denominator.
−2+5=3-2 + 5 = 3
Add −2-2 and 55.
4
Divide the numerator by the denominator.
123=4\frac{12}{3} = 4
Simplify the fraction to get the final integer answer.

Key Concept

Evaluating algebraic expressions requires substituting specific numerical values into the expression and carefully applying the order of operations, especially when handling negative numbers raised to powers.
Question 278Question →

Evaluate the algebraic expression for the given variable values.

Fill in the blanks below

When p=−3p = -3 and q=4q = 4, the value of the expression 2p2−5q+12p^2 - 5q + 1 is .
Show answer & explanation

Answer

The value of the expression is -1.
Substituting p=−3p = -3 and q=4q = 4 into 2p2−5q+12p^2 - 5q + 1 yields 2(−3)2−5(4)+1=2(9)−20+1=18−20+1=−12(-3)^2 - 5(4) + 1 = 2(9) - 20 + 1 = 18 - 20 + 1 = -1.

Step-by-Step Solution

1
Substitute the given values p=−3p = -3 and q=4q = 4 into the expression 2p2−5q+12p^2 - 5q + 1.
2(−3)2−5(4)+12(-3)^2 - 5(4) + 1
Replace each variable with its assigned value.
2
Evaluate the exponent (−3)2(-3)^2.
2(9)−5(4)+12(9) - 5(4) + 1
Exponents must be evaluated before multiplication according to the order of operations.
3
Perform the multiplication operations.
18 - 20 + 1
2×9=182 \times 9 = 18 and 5×4=205 \times 4 = 20.
4
Add and subtract from left to right.
-1
18−20=−218 - 20 = -2, and −2+1=−1-2 + 1 = -1.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:45s
Question 279Question →

If x=−2x = -2 and y=3y = 3, what is the value of the algebraic expression 3x2−y2+4x3x^2 - y^2 + 4x?

Show answer & explanation

Answer: −5-5

Answer

−5-5
Substituting x=−2x = -2 and y=3y = 3 gives 3(−2)2−(3)2+4(−2)3(-2)^2 - (3)^2 + 4(-2). Squaring −2-2 yields 44, so the first term becomes 3(4)=123(4) = 12. Squaring 33 yields 99, so the second term becomes −9-9. Multiplying 44 by −2-2 yields −8-8. Summing these values gives 12−9−8=−512 - 9 - 8 = -5.

Step-by-Step Solution

1
Substitute the given values x=−2x = -2 and y=3y = 3 into the expression 3x2−y2+4x3x^2 - y^2 + 4x.
3(−2)2−(3)2+4(−2)3(-2)^2 - (3)^2 + 4(-2)
Replace each variable with its respective numeric value.
2
Evaluate the exponential terms according to the order of operations (PEMDAS).
3(4)−9+4(−2)3(4) - 9 + 4(-2)
Squaring a negative number yields a positive result: (−2)2=4(-2)^2 = 4, and (3)2=9(3)^2 = 9.
3
Perform multiplication operations.
12−9−812 - 9 - 8
3⋅4=123 \cdot 4 = 12 and 4⋅(−2)=−84 \cdot (-2) = -8.
4
Add and subtract from left to right.
−5-5
12−9=312 - 9 = 3, and 3−8=−53 - 8 = -5.

Key Concept

Evaluating Algebraic Expressions with Signed Numbers
Question 280Question →

If x=−2x = -2, y=12y = \frac{1}{2}, and z=−3z = -3, what is the value of the algebraic expression x3y−2−4z(x+2y)2−z\frac{x^3 y^{-2} - 4z}{(x + 2y)^2 - z}?

Show answer & explanation

Answer: −5-5

Answer

The value of the expression is −5-5.
Substituting x=−2x = -2, y=12y = \frac{1}{2}, and z=−3z = -3 yields a numerator of (−2)3(12)−2−4(−3)=(−8)(4)+12=−20(-2)^3 \left(\frac{1}{2}\right)^{-2} - 4(-3) = (-8)(4) + 12 = -20, and a denominator of (−2+2(12))2−(−3)=(−1)2+3=4\left(-2 + 2\left(\frac{1}{2}\right)\right)^2 - (-3) = (-1)^2 + 3 = 4. Dividing −20-20 by 44 gives the final answer −5-5.

Step-by-Step Solution

1
Evaluate the terms in the numerator: x3y−2−4zx^3 y^{-2} - 4z
Since x=−2x = -2, x3=(−2)3=−8x^3 = (-2)^3 = -8. Since y=12y = \frac{1}{2}, y−2=(12)−2=22=4y^{-2} = \left(\frac{1}{2}\right)^{-2} = 2^2 = 4. So x3y−2=(−8)(4)=−32x^3 y^{-2} = (-8)(4) = -32. Also, −4z=−4(−3)=12-4z = -4(-3) = 12. The numerator simplifies to −32+12=−20-32 + 12 = -20.
Negative exponents indicate reciprocals, and cubing a negative base yields a negative result.
2
Evaluate the terms in the denominator: (x+2y)2−z(x + 2y)^2 - z
Inside the parentheses, x+2y=−2+2(12)=−2+1=−1x + 2y = -2 + 2\left(\frac{1}{2}\right) = -2 + 1 = -1. Squaring this gives (−1)2=1(-1)^2 = 1. Subtracting zz gives 1−(−3)=1+3=41 - (-3) = 1 + 3 = 4.
Operations inside parentheses must be calculated before applying the exponent, and subtracting a negative integer is equivalent to adding its positive.
3
Divide the numerator by the denominator
−204=−5.\frac{-20}{4} = -5.
Dividing a negative integer by a positive integer produces a negative quotient.

Key Concept

Evaluating algebraic expressions involving negative exponents, integer substitutions, and order of operations.
Estimated Time:1m 30s
PreviousPage 14 / 16Next