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ACT Math : Inequalities

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Example Question #1 :How To Find The Solution To An Inequality With Subtraction

Given thatx= 2 andy= 4, how much less is the value of 2x²– 2ythan the value of 2y²– 2x?

Possible Answers:

Correct answer:

Explanation:

First, we solve each expression by plugging in the given values forxandy:

2(2²) – 2(4) = 8 – 8 = 0

2(4²) – 2(2) = 32 – 4 = 28

Then we find the difference between the first and second expressions’ values:

28 – 0 = 28

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Example Question #2 :Inequalities

Solve | z + 5 | > 2

Possible Answers:

z > 5

z < 2

2 < z < 5

-7 < z < -3

$z > -3 \; or\; z < -7$

Correct answer:

$z > -3 \; or\; z < -7$

Explanation:

Absolute value problems are broken into two inequalities: z + 5 > 2 and z + 5 < -2 . Each inequality is solved separately to get z > -3 and z < -7 . Graphing each inequality shows that the correct answer is $z > -3\; OR\; z < -7$ .

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Example Question #3 :Inequalities

Which of the following inequalities defines the solution set to $3x+1\leq-4x-6$ ?

Possible Answers:

$x\leq-1$

$x\leq1$

$x\leq-7$

$x\geq-1$

$x\geq1$

Correct answer:

$x\leq-1$

Explanation:

First, move thes to one side.

$7x+1\leq-6$

Subtract by 1

$7x\leq-7$

Divide both sides by 7.

$x\leq-1$

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Example Question #2 :Inequalities

The cost, in cents, of manufacturing $\dpi{100} \small x$ pencils is $\dpi{100} \small 1200+20x$ , where 1200 is the number of cents required to run the factory regardless of the number of pencils made, and 20 represents the per-unit cost, in cents, of making each pencil. The pencils sell for 50 cents each. What number of pencils would need to be sold so that the revenue received is at least equal to the manufacturing cost?

Possible Answers:

$\dpi{100} \small 40$

$\dpi{100} \small 30$

$\dpi{100} \small 27$

$\dpi{100} \small 36$

$\dpi{100} \small 33$

Correct answer:

$\dpi{100} \small 40$

Explanation:

If each pencil sells at 50 cents, $\dpi{100} \small x$ pencils will sell at $\dpi{100} \small 50x$ . The smallest value of $\dpi{100} \small x$ such that

$\dpi{100} \small 50x\geq 1200+20x$

$\dpi{100} \small x\geq 40$

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Example Question #5 :Inequalities

Solve the following inequality:

x-2<13

Possible Answers:

x<11

x>15

x<15

x>11

Correct answer:

x<15

Explanation:

To solve an inequality with subtraction, simply solve it is as 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.

In this case add two to each side.

x-2<13

x-2=13

x=15

x<15

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Example Question #1 :How To Find The Solution To An Inequality With Subtraction

Solve the following inequality:

3x-7<5

Possible Answers:

x<4

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

x>4

$x>\frac{5}{3}$

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,

$3x-7<5\Rightarrow 3x<12\Rightarrow x<4$

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Example Question #1 :How To Find The Solution To An Inequality With Addition

Solve |x– 5| ≤ 1

Possible Answers:

None of the answers are correct

-1 ≤ x ≤ 1

x ≤ 4 or x ≥ 6

0 ≤ x ≤ 1

4 ≤ x ≤ 6

Correct answer:

4 ≤ x ≤ 6

Explanation:

Absolute values have two answers: a positive one and a negative one. Therefore,

-1 ≤ x – 5≤ 1 and solve by adding 5 to all sides to get 4 ≤ x ≤ 6.

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Example Question #1 :How To Find The Solution To An Inequality With Addition

Solve $|z - 3| \leq 5$

Possible Answers:

All real numbers

No solutions

$z \leq 8$

$-2 \leq z \leq 8$

$-2 \geq z$

Correct answer:

$-2 \leq z \leq 8$

Explanation:

Absolute value is the distance from the origin and is always positive.

So we need to solve $z - 3 \leq 5$ and $z - 3 \geq -5$ which becomes a bounded solution.

Adding 3 to both sides of the inequality gives $z \leq 8$ and $z \geq -2$ or in simplified form $-2 \leq z\leq 8$

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Example Question #2 :Inequalities

Given the inequality $\left | 12-2x\right |>2,$ which of the following is correct?

Possible Answers:

x<-7 or x>5

-5<x<7

x<-5 or x>7

5<x<7

x<5 or x>7

Correct answer:

x<5 or x>7

Explanation:

First separate the inequality $\left | 12-2x\right |>2$ into two equations.

12-2x>2

12-2x<-2

Solve the first inequality.

12-2x>2

-2x>-10

x<5

Solve the second inequality.

12-2x<-2

-2x<-14

x>7

Thus, x<5 or x>7 .

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Example Question #1 :How To Find The Solution To An Inequality With Addition

What values ofxmake the following statement true?

|x– 3| < 9

Possible Answers:

–6 <x< 12

–3 <x< 9

6 <x< 12

x< 12

–12 <x< 6

Correct answer:

–6 <x< 12

Explanation:

Solve the inequality by adding 3 to both sides to getx< 12. Since it is absolute value,x– 3 > –9 must also be solved by adding 3 to both sides so:x> –6 so combined.

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ACT Math : Inequalities

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 :How To Find The Solution To An Inequality With Subtraction

Example Question #2 :Inequalities

Example Question #3 :Inequalities

Example Question #2 :Inequalities

Example Question #5 :Inequalities

Example Question #1 :How To Find The Solution To An Inequality With Subtraction

Example Question #1 :How To Find The Solution To An Inequality With Addition

Example Question #1 :How To Find The Solution To An Inequality With Addition

Example Question #2 :Inequalities

Example Question #1 :How To Find The Solution To An Inequality With Addition

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