Solving Linear Inequalities Practice Questions — ACT — Page 2

Question 21Question →

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

h \leq \frac{16}{5}

h \geq \frac{8}{5}

h \geq \frac{16}{5}

h \leq -\frac{2}{9}

h \geq \frac{1}{2}

Show answer & explanation

Answer: $h \geq \frac{16}{5}$

Answer

h \geq \frac{16}{5}

The correct answer is the inequality stating that

h

must be greater than or equal to sixteen-fifths. This is found by setting up the linear inequality

\frac{7}{2} - \frac{5}{4}h \leq -\frac{1}{2}

, subtracting

\frac{7}{2}

from both sides to get

-\frac{5}{4}h \leq -4

, and then multiplying by

-\frac{4}{5}

while reversing the inequality sign.

Step-by-Step Solution

1

Set up the inequality representing the temperature constraint.

\frac{7}{2} - \frac{5}{4}h \leq -\frac{1}{2}

The initial temperature is

\frac{7}{2}

, the rate of decrease is

\frac{5}{4}

per hour

h

, and the final temperature must be at most

-\frac{1}{2}

.

2

Subtract

\frac{7}{2}

from both sides of the inequality.

-\frac{5}{4}h \leq -4

This isolates the variable term on the left side of the inequality. The subtraction is

-\frac{1}{2} - \frac{7}{2} = -\frac{8}{2} = -4

.

3

Multiply both sides by

-\frac{4}{5}

and reverse the inequality sign.

h \geq \frac{16}{5}

Multiplying or dividing by a negative number requires reversing the direction of the inequality sign. The calculation is

-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 22Question →

For all real values of $x$ that satisfy the inequality $5 - \frac{3x - 2}{4} \ge \frac{x + 6}{2}$ , the solution set is represented by $x \le b$ . What is the value of $b$ ?

Show answer & explanation

Answer: 2

Answer

The value of

b

is 2.

To solve

5 - \frac{3x - 2}{4} \ge \frac{x + 6}{2}

, multiply all terms by 4 to clear the denominators, resulting in

20 - (3x - 2) \ge 2(x + 6)

. Carefully distribute the negative sign to obtain

20 - 3x + 2 \ge 2x + 12

. Simplifying the left side gives

22 - 3x \ge 2x + 12

. Moving the variable terms to one side yields

10 \ge 5x

, which simplifies to

x \le 2

. Thus, the upper bound value

b

is 2.

Step-by-Step Solution

1

Multiply both sides of the inequality by the least common denominator, which is 4.

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 \ge 2x + 12

Distributing the negative sign across

(3x - 2)

changes it to

-3x + 2

. Distributing 2 across

(x + 6)

yields

2x + 12

.

3

Combine like terms on the left side of the inequality.

22 - 3x \ge 2x + 12

Combining the constant terms

20

and

2

simplifies the expression to

22

.

4

Isolate the variable terms by adding

3x

and subtracting 12 from both sides.

10 \ge 5x

Grouping variables on one side and constants on the other allows us to solve for

x

.

5

Divide both sides by 5.

2 \ge x

(or

x \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 - \frac{3}{4}x + \frac{2}{4} \ge \frac{1}{2}x + \frac{6}{2}

. This simplifies to

5.5 - 0.75x \ge 0.5x + 3

. Subtracting

0.5x

and

5.5

from both sides gives

-1.25x \ge -2.5

. Dividing by

-1.25

and reversing the inequality sign gives

x \le 2

.

Estimated Time:1m 30s

Question 23Question →

A chemical solution in a laboratory has an initial volume of $50$ milliliters and evaporates at a rate of $\frac{7}{2}$ milliliters per hour. A second chemical solution has an initial volume of $20$ milliliters and evaporates at a rate of $\frac{5}{4}$ milliliters per hour. After how many hours, $t$ , will the volume of the first solution be at most the volume of the second solution?

t \ge \frac{40}{3}

t \le \frac{40}{3}

t \ge 90

t \le 90

t \le -\frac{40}{3}

Show answer & explanation

Answer: $t \ge \frac{40}{3}$

Answer

The correct solution is the set of values

t \ge \frac{40}{3}

.

Translating the problem statement gives the inequality

50 - \frac{7}{2}t \le 20 - \frac{5}{4}t

. Subtracting 50 from both sides and adding

\frac{5}{4}t

to both sides results in

-\frac{9}{4}t \le -30

. Multiplying both sides by

-\frac{4}{9}

and reversing the inequality sign because of the negative multiplier yields the correct solution,

t \ge \frac{40}{3}

.

Step-by-Step Solution

1

Write the inequality representing the physical situation.

50 - \frac{7}{2}t \le 20 - \frac{5}{4}t

The volume of the first solution after

t

hours is

50 - \frac{7}{2}t

, and the volume of the second solution is

20 - \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.

-\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.

-\frac{9}{4}t \le -30

Converting

-\frac{7}{2}t

to

-\frac{14}{4}t

allows us to add it to

\frac{5}{4}t

, resulting in

-\frac{9}{4}t

.

4

Multiply both sides of the inequality by

-\frac{4}{9}

to solve for

t

, reversing the inequality sign because we are multiplying by a negative number.

t \ge \frac{40}{3}

Multiplying by the reciprocal of the coefficient isolates

t

. 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 24Question →

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

Show answer & explanation

Answer: -2

Answer

-2

Solving the inequality by isolating

x

leads to

x \le -2

, meaning that any value of

x

less than or equal to

-2

satisfies the inequality. Therefore, the maximum possible value is

-2

.

Step-by-Step Solution

1

Multiply both sides of the inequality by 3 to clear the denominator.

2 - 5x \ge 6x + 24

Multiplying by a positive number eliminates the fraction without changing the inequality direction.

2

Subtract

6x

from both sides of the inequality.

2 - 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 \ge 22

This isolates the variable term on the left side.

4

Divide both sides by

-11

and reverse the inequality sign.

x \le -2

Dividing by a negative number requires reversing the direction of the inequality sign. The resulting inequality defines the upper bound for

x

.

Key Concept

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

Question 25Question →

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

x \ge -\frac{3}{2}

x \le -\frac{9}{2}

x \le -\frac{3}{2}

x \ge -\frac{9}{2}

x \le -\frac{25}{4}

Show answer & explanation

Answer: $x \le -\frac{3}{2}$

Answer

The inequality that represents the complete solution set is

x \le -\frac{3}{2}

The correct answer is the inequality stating that

x

is less than or equal to negative three-halves. Simplifying the expression yields

-8x \ge 12

, and dividing by

-8

correctly reverses the inequality direction to produce

x \le -\frac{3}{2}

.

Step-by-Step Solution

1

Distribute

-3

to both terms inside the parentheses on the left side of the inequality.

7 - 3x + 12 \ge 5x + 31

To eliminate the parentheses and prepare the inequality for simplification.

2

Combine the constant terms

7

and

12

on the left side.

19 - 3x \ge 5x + 31

To group like terms on the left side.

3

Subtract

5x

from both sides of the inequality to collect variable terms on one side.

19 - 8x \ge 31

To isolate the variable term on the left side.

4

Subtract

19

from both sides of the inequality.

-8x \ge 12

To isolate the variable term completely before division.

5

Divide both sides of the inequality by

-8

and flip the inequality sign.

x \le -\frac{12}{8}

, which simplifies to

x \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 26Question →

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

Show answer & explanation

Answer: 5

Answer

The smallest integer value of

n

that satisfies the inequality is 5.

Solving the inequality yields

n > \frac{103}{22} \approx 4.68

. The smallest integer greater than

4.68

is

5

.

Step-by-Step Solution

1

Multiply both sides by 20 to clear the denominators.

5(3 - 2n) < 4(3n - 22)

To eliminate the fractions and simplify the inequality.

2

Distribute the constants on both sides.

15 - 10n < 12n - 88

To remove the parentheses.

3

Subtract

12n

from both sides.

15 - 22n < -88

To group the terms containing the variable on the left side.

4

Subtract 15 from both sides.

-22n < -103

To isolate the variable term on the left side.

5

Divide both sides by

-22

and reverse the inequality sign.

n > \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 > 4.68

To find the smallest integer value that satisfies this condition.

7

Identify the smallest integer greater than

4.68

.

5

The smallest integer greater than

4.68

is

5

.

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 27Question →

Which of the following inequalities represents the complete solution set for $x$ in the inequality $\frac{x}{3} - \frac{2x - 5}{2} \ge 2$ ?

x \le \frac{3}{4}

x \ge \frac{3}{4}

x \le 3

x \le -\frac{27}{4}

x \ge 3

Show answer & explanation

Answer: $x \le \frac{3}{4}$

Answer

The inequality is satisfied by all values of

x

less than or equal to

\frac{3}{4}

.

The correct solution is obtained by rewriting the fractions with a common denominator of 6, which yields

\frac{2x - 3(2x - 5)}{6} \ge 2

. Distributing the negative sign gives

\frac{-4x + 15}{6} \ge 2

. Multiplying both sides by 6 and subtracting 15 results in

-4x \ge -3

. Dividing by

-4

and reversing the inequality sign yields

x \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.

\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.

\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 \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 \ge -3

This isolates the variable term on the left side.

5

Divide both sides by

-4

and reverse the direction of the inequality sign.

x \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 \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) \ge 12

, which reduces to

-4x + 15 \ge 12

.

Estimated Time:1m 30s

Question 28Question →

A warehouse is cooling a refrigerated storage room. The initial temperature of the room is $22^\circ\text{C}$ . A cooling system lowers the temperature by $1.8^\circ\text{C}$ per hour. To store a specific vaccine, the temperature of the room must be kept strictly below $-5^\circ\text{C}$ . 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 < -5

. Subtracting 22 from both sides yields

-1.8h < -27

. Dividing by the negative rate

-1.8

requires reversing the inequality sign, which gives

h > 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 < -5

The initial temperature is

22^\circ\text{C}

, and it decreases by

1.8^\circ\text{C}

per hour

h

. The temperature must be strictly below

-5^\circ\text{C}

.

2

Isolate the variable term by subtracting 22 from both sides.

-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 > 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 29Question →

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

x < -\frac{15}{8}

x > -\frac{15}{8}

x < \frac{1}{2}

x > \frac{1}{2}

x > 24

Show answer & explanation

Answer: $x < -\frac{15}{8}$

Answer

x < -\frac{15}{8}

Subtracting

\frac{3}{4}

from both sides of the inequality gives

-\frac{2x}{3} > \frac{5}{4}

. Multiplying both sides by

-\frac{3}{2}

and reversing the inequality sign from greater than (

>

) to less than (

<

) yields

x < -\frac{15}{8}

.

Step-by-Step Solution

1

Subtract

\frac{3}{4}

from both sides of the inequality.

-\frac{2x}{3} > \frac{5}{4}

To isolate the term containing the variable

x

on one side of the inequality.

2

Multiply both sides of the inequality by

-\frac{3}{2}

and reverse the inequality sign.

x < -\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 30Question →

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

Show answer & explanation

Answer: -4

Answer

The greatest integer value of

x

that satisfies the inequality is

-4

.

To find the greatest integer value of

x

that satisfies the inequality, solve the inequality algebraically. First, clear the fraction by multiplying all terms by 3:

7 - 5x - 12 \ge 6(x + 6)

. Simplify the left side to

-5x - 5

and distribute the right side to get

6x + 36

. Move variables to the left side by subtracting

6x

to get

-11x - 5 \ge 36

. Add 5 to both sides to get

-11x \ge 41

. Finally, divide by

-11

and reverse the inequality sign, yielding

x \le -\frac{41}{11}

, which is approximately

x \le -3.73

. The greatest integer less than or equal to

-3.73

is

-4

.

Step-by-Step Solution

1

Multiply the entire inequality by 3 to clear the fraction

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 \ge 6x + 36

Combines constant terms on the left side and expands the parentheses on the right side.

3

Subtract

6x

from both sides

-11x - 5 \ge 36

Gathers all variable terms on the left side of the inequality.

4

Add 5 to both sides

-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 \le -\frac{41}{11}

Isolates the variable

x

. 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 \le -3.73

, so the greatest integer is

-4

Since

-\frac{41}{11} \approx -3.73

, the largest integer that is less than or equal to this value is

-4

.

Key Concept

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

Solving Linear Inequalities

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

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Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

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Step-by-Step Solution

Key Concept

Common Mistakes

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Step-by-Step Solution

Key Concept

Common Mistakes