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ACT Math : Inequalities

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Example Questions

Example Question #31 : Inequalities

Solve the following inequality:

$-\frac{1}{3}x>3$

Possible Answers:

x>-1

x<-1

x<-9

x>-9

Correct answer:

x<-9

Explanation:

To solve simply solve as though it is an equation.

The goal is to isolate the variable on one side with all other constants on the other side. Perform the opposite operation to manipulate the inequality.

However, remember that when dividing or multiplying by a negative number, you must flip the inequality sign.

$-\frac{1}{3}x>3$

Multiply by -3, thus:

x<-9

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Example Question #32 : Inequalities

Solve the following inequality:

-4x>16

Possible Answers:

x>-4

x<4

x>4

x<-4

Correct answer:

x<-4

Explanation:

To solve, simply treat it as an equation.

This means you want to isolate the variable on one side and move all other constants to the other side through opposite operation manipulation.

Remember, you only flip the inequality sign if you multiply or divide by a negative number.

Thus,

$-4x>16\Rightarrow \frac{-4x}{-4}>\frac{16}{-4}\Rightarrow x<-4$

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Example Question #33 : Inequalities

Which of the following inequalities defines the solution set for the inequality 14 – 3x ≤ 5?

Possible Answers:

x ≥ 3

x ≤ 3

x ≤ –19/3

x ≥ –3

x ≤ –3

Correct answer:

x ≥ 3

Explanation:

To solve this inequality, you should first subtract 14 from both sides.

This leaves you with –3x ≤ –9.

In the next step, you divide both sides by –3, remembering to flip the inequality sign when you do this.

This leaves you with the solution x ≥ 3.

If you selected x ≤ 3, you probably forgot to flip the sign. If you selected one of the other solutions, you may have subtracted incorrectly.

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Example Question #2 : How To Find The Solution To An Inequality With Division

Solve 6x – 13 > 41

Possible Answers:

x < 6

x > 6

x < 9

x > 4.5

x > 9

Correct answer:

x > 9

Explanation:

Add 13 to both sides, giving you 6x > 54, divide both sides by 6, leaving x > 9.

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Example Question #34 : Inequalities

Solve for .

$\small 14-2x\geq 22$

Possible Answers:

$\small x\geq-4$

$\small x\leq-4$

$\small x\leq4$

$\small x\geq4$

Correct answer:

$\small x\leq-4$

Explanation:

$\small 14-2x\geq22$

$\small -2x\geq8$

When dividing both sides of an inequality by a negative number, you must change the direction of the inequality sign.

$\small x\leq-4$

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Example Question #35 : Inequalities

Solve 3 < 5x + 7

Possible Answers:

2 > x

$-\frac{3}{5}<x$

$-\frac{4}{5}>x$

2<x

$-\frac{4}{5}<x$

Correct answer:

$-\frac{4}{5}<x$

Explanation:

Subtract seven from both sides, then divide both sides by 5, giving you –4/5 < x.

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Example Question #36 : Inequalities

Find is the solution set for x where:

$\left | 5x-3\right |>12$

Possible Answers:

x> 3 or $x< -\frac{9}{5}$

$-3< x< \frac{9}{5}$

$-\frac{9}{5}< x< 3$

x> -3 or $x< \frac{9}{5}$

$x< -\frac{9}{5}$

Correct answer:

x> 3 or $x< -\frac{9}{5}$

Explanation:

We start by splitting this into two inequalities, (5x-3)>12 and (5x-3)<-12

We solve each one, giving us x>3 or $x<\frac{-9}{5}\right$ .

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Example Question #37 : Inequalities

Which of the following could be a value of , given the following inequality?

$8-7x\leq11x+4$

Possible Answers:

$\frac{1}{9}$

$-\frac{1}{3}$

$-\frac{2}{9}$

$-\frac{2}{3}$

$\frac{4}{9}$

Correct answer:

$\frac{4}{9}$

Explanation:

The inequality that is presented in the problem is:

$8-7x\leq11x+4$

Start by moving your variables to one side of the inequality and all other numbers to the other side:

$-7x-11x\leq4-8$

$-18x\leq-4$

Divide both sides of the equation by . Remember to flip the direction of the inequality's sign since you are dividing by a negative number!

$x\geq\frac{-4}{-18}$

Reduce:

$x\geq\frac{2}{9}$

The only answer choice with a value greater than $\frac{2}{9}$ is $\frac{4}{9}$ .

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Example Question #165 : Equations / Inequalities

Solve for the -intercept:

$3y+11\geq 5y+6x-1$

Possible Answers:

Correct answer:

Explanation:

Don't forget to switch the inequality direction if you multiply or divide by a negative.

$3y+11\geq 5y+6x-1$

$-2y+11\geq6x-1$

$-2y\geq6x-12$

$-\frac{1}{2}(-2y\geq 6x-12)$

$y\leq -3x+6$

Now that we have the equation in slope-intercept form, we can see that the y-intercept is 6.

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Example Question #5 : How To Find The Solution To An Inequality With Division

Solve for :

-4x - 12 < 15

Possible Answers:

$x < \frac{27}{4}$

$x<\frac{15}{4}$

$x > \frac{15}{4}$

x < 15-4x

$x > -\frac{27}{4}$

Correct answer:

$x > -\frac{27}{4}$

Explanation:

Begin by adding to both sides, this will get the variable isolated:

-4x < 15 + 12

Or...

-4x < 27

Next, divide both sides by :

$x > -\frac{27}{4}$

Notice that when you divide by a negative number, you need to flip the inequality sign!

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ACT Math : Inequalities

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #31 : Inequalities

Example Question #32 : Inequalities

Example Question #33 : Inequalities

Example Question #2 : How To Find The Solution To An Inequality With Division

Example Question #34 : Inequalities

Example Question #35 : Inequalities

Example Question #36 : Inequalities

Example Question #37 : Inequalities

Example Question #165 : Equations / Inequalities

Example Question #5 : How To Find The Solution To An Inequality With Division

All ACT Math Resources

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