ACT Math : How to find the solution to an inequality with division

Study concepts, example questions & explanations for ACT Math

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Example Questions

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Example Question #31 : Inequalities

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

Possible Answers:

x ≤ –19/3

x ≥ –3

x ≤ 3

x ≥ 3

x ≤ –3

Correct answer:

x ≥ 3

Explanation:

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

This leaves you with –3x ≤ –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 #1 : How To Find The Solution To An Inequality With Division

Solve 6x – 13 > 41

Possible Answers:

< 6

> 6

< 9

> 9

> 4.5

Correct answer:

> 9

Explanation:

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

Example Question #33 : Inequalities

Solve for .

\small 14-2x\geq 22

Possible Answers:

\small x\leq-4

\small x\geq-4

\small x\geq4

\small x\leq4

Correct answer:

\small x\leq-4

Explanation:

\small 14-2x\geq22

\small -2x\geq8

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

\small x\leq-4

Example Question #34 : Inequalities

Solve 3 < 5x + 7

Possible Answers:

2 > x

Correct answer:

Explanation:

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

Example Question #1 : 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 

Explanation:

We start by splitting this into two inequalities,  and  

We solve each one, giving us  or .

Example Question #2 : 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:

Explanation:

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 #1 : Inequalities

Solve for the -intercept:

3y+11\geq 5y+6x-1

Possible Answers:

Correct answer:

Explanation:

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

3y+11\geq 5y+6x-1

-2y+11\geq6x-1

-2y\geq6x-12

-\frac{1}{2}(-2y\geq 6x-12)

y\leq -3x+6

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

Example Question #3 : Inequalities

Solve for :

Possible Answers:

Correct answer:

Explanation:

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 #41 : Inequalities

Each of the following is equivalent to  

xy/z * (5(x + y))  EXCEPT:

 

Possible Answers:

xy(5y + 5x)/z

5x²y + 5xy²/z

5x² + y²/z

xy(5x + 5y)/z

Correct answer:

5x² + y²/z

Explanation:

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 \dpi{100} \small 3x+8<35 ?

Possible Answers:

\dpi{100} \small x<35

\dpi{100} \small x>27

\dpi{100} \small x>9

\dpi{100} \small x<9

\dpi{100} \small x<27

Correct answer:

\dpi{100} \small x<9

Explanation:

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

\dpi{100} \small 3x+8-8<35-8

\dpi{100} \small \frac{3x}{3}<\frac{27}{3}

Thus \dpi{100} \small x<9

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