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Algebra II : Solving Inequalities

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Example Questions

Example Question #1 : Solving Inequalities

Solve for .

$-7x-4\geq10$

Possible Answers:

$x\geq2$

$x\leq-2$

$x\leq2$

$x\geq-2$

Correct answer:

$x\leq-2$

Explanation:

$-7x-4\geq10$

Add 4 to both sides.

$-7x\geq14$

Divide both sides by –7. When dividing by a negative value, we must also change the direction of the inequality sign.

$x\leq-2$

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Example Question #1 : Solving Inequalities

Solve this inequality.

$\left | \frac{4x+3}{x+6}\right |\eq>1$

Possible Answers:

$x\eq<\frac{9}{5}, x\eq>-1$

$x<-\frac{9}{5}, x\eq<1$

$x \eq< -\frac{9}{5}, \hspace x \eq>1$

$x\eq>-\frac{9}{5}, x\eq>1$

$x\eq>-\frac{9}{5}, x \eq<1$

Correct answer:

$x \eq< -\frac{9}{5}, \hspace x \eq>1$

Explanation:

Split the inequality into two possible cases as follows, based on the absolute values.

First case: $\frac{4x+3}{x+6}\eq>1$

Second case: $\frac{4x+3}{x+6} \eq< -1$

Let's find the inequality of the first case.

$\frac{4x+3}{x+6} \eq>1$

Multiply both sides by x + 6.

$4x+3 \eq>x+6$

Subtract x from both sides, then subtract 3 from both sides.

$3x \eq>3$

Divide both sides by 3.

$x\eq>1$

Let's find the inequality of the second case.

$\frac{4x+3}{x+6} \eq< -1$

Multiply both sides by x + 6.

$4x+3 \eq< -1(x+6)$

Simplify.

4x+3< -x-6

Add x to both sides, then subtract 3 from both sides.

$5x\eq< -9$

Divide both sides by 5.

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

So the range of x-values is $x<-\frac{9}{5}$ and $x \eq>1$ .

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Example Question #2 : Solving Inequalities

Solve the compound inequality and express answer in interval notation:

$\small x-3 > -1$ or $\small x+2 < 4$

Possible Answers:

$\small (-\infty , 2) \cup \left (2, \infty \right )$

$\small \varnothing$ (no solution)

$\small \left (-\infty, \infty \right )$

$\small (2,2)$

Correct answer:

$\small (-\infty , 2) \cup \left (2, \infty \right )$

Explanation:

For a compound inequality, we solve each inequality individually. Thus, for the first inequality, $\small x-3 > -1$ , we obtain the solution x>2 and for the second inequality, $\small x+2 < 4$ , we obtain the solution x<2 . In interval notation, the solutions are $\small (2, \infty)$ and $\small (-\infty, 2)$ , respectively. Because our compound inequality has the word "or", this means we union the two solutions to obatin $\small (-\infty , 2) \cup \left (2, \infty \right )$ .

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Example Question #2 : Writing Inequalities

Find the solution set of the inequality:

$6x + 22 \leq 97$

Possible Answers:

$[12.5, \infty)$

$[-12.5, \infty)$

$(-\infty , 12.5]$

$\left ( -\infty , \infty \right )$

$(-\infty , -12.5]$

Correct answer:

$(-\infty , 12.5]$

Explanation:

$6x + 22 \leq 97$

$6x + 22 -22 \leq 97-22$

$6x \leq 75$

$6x \div 6 \leq 75\div 6$

$x \leq 12.5$

or, in interval notation, $(-\infty , 12.5]$

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Example Question #1 : Solving Inequalities

Sam's age is three years more than twice his brother's age. If the sum of their ages is at least 18, then was is the maximum possible age of Sam's brother?

Possible Answers:

years old

Correct answer:

years old

Explanation:

Let be Sam's age, and let be his brother's age.

In the problem, we are told that the sum of their ages is at least 18. Represent this with an inequality:

$x+y\leq 18$

Sam's age is three years more than twice his brothers age. Write this mathematically as:

2y+3=x

Plug in 2y+3 for the value in the inequality and solve for :

$2y+3+y\leq 18$

$3y+3\leq 18$

$3y\leq 15$

$y\leq 5$

The age of Sam's brother is less than or equal to years.

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Example Question #1 : Solving Inequalities

Solve the inequality:

$\small x^2+4< 9$

Possible Answers:

$\small x< -\sqrt5$

$\small -\sqrt5< x< \sqrt5$

$\small -\sqrt5> x\ or\ x> \sqrt5$

$\small x< \sqrt5$

Correct answer:

$\small -\sqrt5< x< \sqrt5$

Explanation:

$\small \small x^2+4< 9$

$\small x^2< 5$

$\small x< \left | (\pm\sqrt5) \right |$

$\small -\sqrt5< x< \sqrt5$

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Example Question #5 : Solving Inequalities

Solve the following inequality for :

$\large -7x-2 \ge 12$

Possible Answers:

$\large x \ge 2$

$\large x \ge -2$

x > -10/7

$x \le -2$

Correct answer:

$x \le -2$

Explanation:

Most of solving inequalities is straightforward algebra and we can manipulate them in the same way as equations in most cases.

$\large -7x-2 \ge 12$

$\large -7x \ge 14$

However, we must remember that when multiplying or dividing by negative numbers in inequalities, we have to switch the direction of the inequality. So we do the final division step and get the answer:

$\large x \le -2$

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Example Question #2 : Solving Inequalities

Solve for :

$3x+20\leq -5x-4$

Possible Answers:

$x \leq 3$

$x \leq -3$

$x \geq 3$

x < -3

$x \geq -3$

Correct answer:

$x \leq -3$

Explanation:

In order to solve this inequality, we must first consolidate all of our values on one side.

The first thing we need to do is move the to the other side:

$3x+\underset{-20}{20} \le -5x - \underset{-20}{4}$

This results in:

$3x \le -5x-24$

Next, we need to move the -5x from the right side over to the left side:

$\underset{+5x}{3x} \le \underset{+5x}{-5x}-24$

This gives us

$8x \le -24$

Dividing each side by gives us our solution:

$x \le -3$

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Example Question #7 : Solving Inequalities

What are the possible values of if 2x+7>5 and 3x-7<2x ?

Possible Answers:

-1<x<7

0<x<9

x=8

x<7

Correct answer:

-1<x<7

Explanation:

The two equations should be solved separately to get,

2x+7>5

2x>-2

x>-1 and

3x-7<2x

x<7 .

This can be checked by plugging in values between and and seeing if they satisfy both equations.

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Example Question #1 : Writing Inequalities

Find the solution set of the inequality:

6x + 22 > 67

Possible Answers:

$\left ( - \infty, -7.5 \right )$

$\left ( -7.5, \infty \right )$

$\left ( -\infty , \infty \right )$

$\left ( - \infty, 7.5 \right )$

$\left ( 7.5, \infty \right )$

Correct answer:

$\left ( 7.5, \infty \right )$

Explanation:

6x + 22 > 67

6x + 22 -22 > 67 -22

6x > 45

$6x \div 6 > 45\div 6$

x > 7.5

or, in interval notation, $\left ( 7.5, \infty \right )$

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Algebra II : Solving Inequalities

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Solving Inequalities

Example Question #1 : Solving Inequalities

Example Question #2 : Solving Inequalities

Example Question #2 : Writing Inequalities

Example Question #1 : Solving Inequalities

Example Question #1 : Solving Inequalities

Example Question #5 : Solving Inequalities

Example Question #2 : Solving Inequalities

Example Question #7 : Solving Inequalities

Example Question #1 : Writing Inequalities

All Algebra II Resources

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