# ACT Math : Inequalities

## Example Questions

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### Example Question #1 : Inequalities

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

52

12

28

2

28

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

### Example Question #2 : Inequalities

Solve

Explanation:

Absolute value problems are broken into two inequalities:   and .  Each inequality is solved separately to get  and .  Graphing each inequality shows that the correct answer is .

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

Which of the following inequalities defines the solution set to  ?

Explanation:

First, move the s to one side.

Subtract by 1

Divide both sides by 7.

### Example Question #10 : Inequalities

The cost, in cents, of manufacturing  pencils is , 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?

Explanation:

If each pencil sells at 50 cents,  pencils will sell at . The smallest value of  such that

### Example Question #4 : Inequalities

Solve the following inequality:

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.

### Example Question #1 : Inequalities

Solve the following inequality:

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,

### Example Question #1 : Inequalities

Solve |x – 5| ≤ 1

0 ≤ x ≤ 1

None of the answers are correct

-1 ≤ x ≤ 1

4 ≤ x ≤ 6

x ≤ 4 or x ≥ 6

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.

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

Solve

All real numbers

No solutions

Explanation:

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

So we need to solve  and  which becomes a bounded solution.

Adding 3 to both sides of the inequality gives  and  or in simplified form

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

Given the inequality  which of the following is correct?

or

or

or

or

Explanation:

First separate the inequality   into two equations.

Solve the first inequality.

Solve the second inequality.

Thus,   or .

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

What values of x make the following statement true?

|x – 3| < 9

x < 12

–3 < x < 9

–12 < x < 6

6 < x < 12

–6 < x < 12