### All ACT Math Resources

## Example Questions

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

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

**Possible Answers:**

*x* ≤ –19/3

*x* ≤ –3

*x* ≥ –3

*x* ≤ 3

*x* ≥ 3

**Correct answer:**

*x* ≥ 3

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

This leaves you with –3*x* ≤ –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.

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

Solve 6*x* – 13 > 41

**Possible Answers:**

*x *> 4.5

*x *< 6

*x *< 9

*x *> 9

*x *> 6

**Correct answer:**

*x *> 9

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

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

Solve for .

**Possible Answers:**

**Correct answer:**

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

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

Solve 3 < 5x + 7

**Possible Answers:**

2 > x

**Correct answer:**

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

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

Find is the solution set for x where:

**Possible Answers:**

or

or

**Correct answer:**

or

We start by splitting this into two inequalities, and

We solve each one, giving us or .

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

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

**Possible Answers:**

**Correct answer:**

The inequality that is presented in the problem is:

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

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!

Reduce:

The only answer choice with a value greater than is .

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

Solve for the -intercept:

**Possible Answers:**

**Correct answer:**

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

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

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

Solve for :

**Possible Answers:**

**Correct answer:**

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

Or...

Next, divide both sides by :

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

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

Each of the following is equivalent to

xy/z * (5(x + y)) EXCEPT:

**Possible Answers:**

xy(5y + 5x)/z

5x² + y²/z

xy(5x + 5y)/z

5x²y + 5xy²/z

**Correct answer:**

5x² + y²/z

Choice a is equivalent because we can say that technically we are multiplying two fractions together: (xy)/z and (5(x + y))/1. We multiply the numerators together and the denominators together and end up with xy (5x + 5y)/z. xy (5y + 5x)/z is also equivalent because it is only simplifying what is inside the parentheses and switching the order- the commutative property tells us this is still the same expression. 5x²y + 5xy²/z is equivalent as it is just a simplified version when the numerators are multiplied out. Choice 5x² + y²/z is not equivalent because it does not account for all the variables that were in the given expression and it does not use FOIL correctly.

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

What is the solution set of the inequality ?

**Possible Answers:**

**Correct answer:**

We simplify this inequality similarly to how we would simplify an equation

Thus

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