### All ACT Math Resources

## Example Questions

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

Solve |*x* – 5| ≤ 1

**Possible Answers:**

-1 ≤ x ≤ 1

x ≤ 4 or x ≥ 6

0 ≤ x ≤ 1

4 ≤ x ≤ 6

None of the answers are correct

**Correct answer:**

4 ≤ x ≤ 6

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 : Inequalities

Solve

**Possible Answers:**

All real numbers

No solutions

**Correct answer:**

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 #2 : Inequalities

Given the inequality which of the following is correct?

**Possible Answers:**

or

or

or

**Correct answer:**

or

First separate the inequality into two equations.

Solve the first inequality.

Solve the second inequality.

Thus, or .

### Example Question #2 : Inequalities

What values of *x* make the following statement true?

|*x* – 3| < 9

**Possible Answers:**

6 < *x* < 12

–3 < *x* < 9

*x* < 12

–6 < *x* < 12

–12 < *x* < 6

**Correct answer:**

–6 < *x* < 12

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

### Example Question #11 : Inequalities

If –1 < *w* < 1, all of the following must also be greater than –1 and less than 1 EXCEPT for which choice?

**Possible Answers:**

3*w*/2

|*w*|^{0.5}

*w*/2

|*w*|

*w*^{2}

**Correct answer:**

3*w*/2

3*w*/2 will become greater than 1 as soon as *w* is greater than two thirds. It will likewise become less than –1 as soon as *w* is less than negative two thirds. All the other options always return values between –1 and 1.

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

Solve for .

**Possible Answers:**

**Correct answer:**

Absolute value problems always have two sides: one positive and one negative.

First, take the problem as is and drop the absolute value signs for the positive side: z – 3 ≥ 5. When the original inequality is multiplied by –1 we get z – 3 ≤ –5.

Solve each inequality separately to get z ≤ –2 or z ≥ 8 (the inequality sign flips when multiplying or dividing by a negative number).

We can verify the solution by substituting in 0 for z to see if we get a true or false statement. Since –3 ≥ 5 is always false we know we want the two outside inequalities, rather than their intersection.

### Example Question #21 : Inequalities

If and , then which of the following could be the value of ?

**Possible Answers:**

**Correct answer:**

To solve this problem, add the two equations together:

The only answer choice that satisfies this equation is 0, because 0 is less than 4.

### Example Question #31 : Inequalities

What values of make the statement true?

**Possible Answers:**

**Correct answer:**

First, solve the inequality :

Since we are dealing with absolute value, must also be true; therefore:

### Example Question #11 : Inequalities

Simplify the following inequality

.

**Possible Answers:**

**Correct answer:**

For a combined inequality like this, you just need to be careful to perform your operations on all the parts of the inequality. Thus, begin by subtracting from each member:

Next, divide all of the members by :

### Example Question #12 : Inequalities

Simplify

.

**Possible Answers:**

**Correct answer:**

Simplifying an inequality like this is very simple. You merely need to treat it like an equation—just don't forget to keep the inequality sign.

First, subtract from both sides:

Then, divide by :

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