Algebra II : Solving Inequalities

Study concepts, example questions & explanations for Algebra II

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Example Questions

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Example Question #1 : Solving Inequalities

Solve for .

Possible Answers:

Correct answer:

Explanation:

Add 4 to both sides.

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

Example Question #2 : Solving Inequalities

Solve this inequality.

Possible Answers:

Correct answer:

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 #3 : Graphing Inequalities

Solve the compound inequality and express answer in interval notation:

   or   

Possible Answers:

 (no solution)

Correct answer:

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 #4 : Writing Inequalities

Find the solution set of the inequality:

Possible Answers:

Correct answer:

Explanation:

or, in interval notation, 

Example Question #3 : 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

years old

years old

years old

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:

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 #4 : Solving Inequalities

Solve the inequality:

Possible Answers:

Correct answer:

Explanation:

Example Question #5 : Solving Inequalities

Solve for m.

Possible Answers:

 

 

Correct answer:

Explanation:

Remember: Use inverse operations to undo the operations in the inequality (for example use a subtraction to undo an addition) until you are left with the variable. Make sure to do the same operations to both sides of the inequality. 

Important Note: When multiplying or dividing by a negative number, always flip the sign of an inequality.

Solution:

Expand all factors

Simplify 

Add 23

 

Subtract 22m

 

Divide by -6 (We flip the sign of the inequality)

Simplify

Example Question #6 : Solving Inequalities

Solve the doulbe inequality and give the solution in interval notation.

Possible Answers:

Correct answer:

Explanation:

Start by subtracting 1 and divinding by 4 on both sides of the equality

Written in interval notation:

Example Question #7 : Solving Inequalities

Solve the following inequality for :

Possible Answers:

Correct answer:

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 #8 : Solving Inequalities

Solve for :

Possible Answers:

Correct answer:

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:

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