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

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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 ≤ 3

x ≤ –19/3

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.

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

Solve 6x – 13 > 41

Possible Answers:

x > 4.5

x < 6

x > 9

x > 6

x < 9

Correct answer:

x > 9

Explanation:

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

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

Solve for .

$\small 14-2x\geq 22$

Possible Answers:

$\small x\leq4$

$\small x\leq-4$

$\small x\geq-4$

$\small x\geq4$

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.

$\small x\leq-4$

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Example Question #1941 : Act Math

Solve 3 < 5x + 7

Possible Answers:

$-\frac{4}{5}<x$

2 > x

2<x

$-\frac{3}{5}<x$

$-\frac{4}{5}>x$

Correct answer:

$-\frac{4}{5}<x$

Explanation:

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

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Example Question #31 : Inequalities

Find is the solution set for x where:

$\left | 5x-3\right |>12$

Possible Answers:

x> 3 or $x< -\frac{9}{5}$

$-3< x< \frac{9}{5}$

$-\frac{9}{5}< x< 3$

x> -3 or $x< \frac{9}{5}$

$x< -\frac{9}{5}$

Correct answer:

x> 3 or $x< -\frac{9}{5}$

Explanation:

We start by splitting this into two inequalities, (5x-3)>12 and (5x-3)<-12

We solve each one, giving us x>3 or $x<\frac{-9}{5}\right$ .

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

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

$8-7x\leq11x+4$

Possible Answers:

$-\frac{1}{3}$

$\frac{1}{9}$

$-\frac{2}{3}$

$\frac{4}{9}$

$-\frac{2}{9}$

Correct answer:

$\frac{4}{9}$

Explanation:

The inequality that is presented in the problem is:

$8-7x\leq11x+4$

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

$-7x-11x\leq4-8$

$-18x\leq-4$

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!

$x\geq\frac{-4}{-18}$

Reduce:

$x\geq\frac{2}{9}$

The only answer choice with a value greater than $\frac{2}{9}$ is $\frac{4}{9}$ .

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

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.

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Example Question #1 : Inequalities

Solve for :

-4x - 12 < 15

Possible Answers:

$x > -\frac{27}{4}$

$x > \frac{15}{4}$

$x<\frac{15}{4}$

$x < \frac{27}{4}$

x < 15-4x

Correct answer:

$x > -\frac{27}{4}$

Explanation:

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

-4x < 15 + 12

Or...

-4x < 27

Next, divide both sides by :

$x > -\frac{27}{4}$

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

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Example Question #1 : 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

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.

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Example Question #7 : Inequalities

What is the solution set of the inequality $\dpi{100} \small 3x+8<35$ ?

Possible Answers:

$\dpi{100} \small x<35$

$\dpi{100} \small x<27$

$\dpi{100} \small x<9$

$\dpi{100} \small x>27$

$\dpi{100} \small x>9$

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

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #31 : Inequalities

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

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

Example Question #1941 : Act Math

Example Question #31 : Inequalities

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

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

Example Question #1 : Inequalities

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

Example Question #7 : Inequalities

All ACT Math Resources

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