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ACT Math : Equations / Inequalities

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Example Questions

Example Question #4 : How To Find Out When An Equation Has No Solution

Solve:

3(2x - 6) + 2x = 7x - 12

Possible Answers:

Correct answer:

Explanation:

First, distribute, making sure to watch for negatives.

3(2x - 6) + 2x = 7x - 12

6x - 18 + 2x = 7x - 12

Combine like terms.

8x - 18 = 7x - 12

Subtract 7x from both sides.

x - 18 = -12

Add 18 on both sides and be careful adding integers.

x = 6

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Example Question #1 : How To Find Out When An Equation Has No Solution

Solve:

-3(2x - 5) = 9 - 6x

Possible Answers:

$\frac{15}{9}$

Infinitely Many Solutions

No Solution

Correct answer:

No Solution

Explanation:

First, distribute the to the terms inside the parentheses.

-3(2x - 5) = 9 - 6x

-6x + 15 = 9 - 6x

Add 6x to both sides.

15 = 9

This is false for any value of . Thus, there is no solution.

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Example Question #2 : How To Find Out When An Equation Has No Solution

Solve $\left | 3-4x\right |<0$ .

Possible Answers:

$x<\frac{4}{3}$

$x>\frac{3}{4}$

$x<\frac{3}{4}$

No solutions

$x>\frac{4}{3}$

Correct answer:

No solutions

Explanation:

By definition, the absolute value of an expression can never be less than 0. Therefore, there are no solutions to the above expression.

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

Given that x = 2 and y = 4, how much less is the value of 2x² – 2y than the value of 2y² – 2x ?

Possible Answers:

Correct answer:

Explanation:

First, we solve each expression by plugging in the given values for x and y:

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 #1 : Inequalities

Solve | z + 5 | > 2

Possible Answers:

2 < z < 5

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

z > 5

-7 < z < -3

z < 2

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 : How To Find The Solution To An Inequality With Subtraction

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

Possible Answers:

$x\leq-7$

$x\geq1$

$x\geq-1$

$x\leq-1$

$x\leq1$

Correct answer:

$x\leq-1$

Explanation:

First, move the s 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 #4 : How To Find The Solution To An Inequality With Subtraction

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:

$\dpi{100} \small 33$

$\dpi{100} \small 36$

$\dpi{100} \small 30$

$\dpi{100} \small 40$

$\dpi{100} \small 27$

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 #4 : How To Find The Solution To An Inequality With Subtraction

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 #4 : Inequalities

Solve |x – 5| ≤ 1

Possible Answers:

-1 ≤ x ≤ 1

x ≤ 4 or x ≥ 6

None of the answers are correct

4 ≤ x ≤ 6

0 ≤ x ≤ 1

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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ACT Math : Equations / Inequalities

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #4 : How To Find Out When An Equation Has No Solution

Example Question #1 : How To Find Out When An Equation Has No Solution

Example Question #2 : How To Find Out When An Equation Has No Solution

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

Example Question #1 : Inequalities

Example Question #3 : How To Find The Solution To An Inequality With Subtraction

Example Question #4 : How To Find The Solution To An Inequality With Subtraction

Example Question #4 : How To Find The Solution To An Inequality With Subtraction

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

Example Question #4 : Inequalities

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