# Algebra II : Solving Inequalities

## Example Questions

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### Example Question #5 : Solving Inequalities

Solve for .

Explanation:

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

### Example Question #1 : Solving Inequalities

Solve this inequality.

Explanation:

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

First case:

Second case:

Let's find the inequality of the first case.

Multiply both sides by x + 6.

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

Divide both sides by 3.

Let's find the inequality of the second case.

Multiply both sides by x + 6.

Simplify.

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

Divide both sides by 5.

So the range of x-values is   and  .

### Example Question #1 : Solving Inequalities

Solve the compound inequality and express answer in interval notation:

or

(no solution)

Explanation:

For a compound inequality, we solve each inequality individually. Thus, for the first inequality, , we obtain the solution  and for the second inequality, , we obtain the solution . In interval notation, the solutions are  and , respectively. Because our compound inequality has the word "or", this means we union the two solutions to obatin

### Example Question #7 : Writing Inequalities

Find the solution set of the inequality:

Explanation:

or, in interval notation,

### 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?

years old

years old

years old

years old

years old

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:

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

Plug in for the value in the inequality and solve for :

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

### Example Question #2 : Solving Inequalities

Solve the inequality:

Explanation:

### Example Question #3 : Solving Inequalities

Solve the following inequality for :

Explanation:

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

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:

### Example Question #4 : Solving Inequalities

Solve for :

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:

This results in:

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

This gives us

Dividing each side by  gives us our solution:

### Example Question #5 : Solving Inequalities

What are the possible values of  if  and  ?

Explanation:

The two equations should be solved separately to get,

and

.

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

### Example Question #8 : Writing Inequalities

Find the solution set of the inequality: