Algebra II : Simplifying Inequalities

Example Questions

Example Question #1 : Simplifying Inequalities

Consider the inequality

Which of the following statements has the same solution set?

Explanation:

can be written as the compound inequality statement

In each expression, 9 can be subtracted to yield the inequality statement

This is the correct response.

Example Question #2 : Simplifying Inequalities

Consider the inequality

Which of the following statements has the same solution set?

Explanation:

is equivalent to the three-way inequality

Add 7 to all three expressions to yield the statement:

This is the correct response.

Example Question #3 : Simplifying Inequalities

Which coordinate is a solution to the inequality

Explanation:

Subtract 1 on both sides of the inequality to get

From here we plug in each set of coordinates to see which could satisfy the statement.

This is a true statement therefore we say that (0,2) satisfies the inequality.

Example Question #4 : Simplifying Inequalities

Simplify the following inequality:

Explanation:

To simplify an inequality we want to isolate the variable on one side of the inequality sign. In order to accomplish this remember to do the reverse opperation to move numbers from one side to the other.

The reverse operation of subtraction is addition. Therefore to move the three from the left hand side to the right hand side we will need to add 3 on each side of the inequality to get

.

From here divide each side of the inequality by 3 to isolate the variable, and since 3>0 we get

.

Example Question #5 : Simplifying Inequalities

Solve the inequality.

.

Explanation:

First, combine the x's on the left side: .

Then, move all the x's to the left side: , which simplifies to .

Next, move all other numbers to the right side: , which simplifies to

Lastly, we divide the entire problem by -1, and this flips the inequality sign.

This gives us .

Now we take the square root of both sides to get our final answer:

Example Question #6 : Simplifying Inequalities

Simplify the following inequality:

Explanation:

First distribute the -4 to give:

Subtract x from both sides to get x on one side to give:

Subtract 8 from both sides to get x term by itself to give:

Divide both sides by -9, and remember when dividing or multiplying both sides by a negative it changes the inequality to give:

Simplify fraction to give final answer:

Example Question #7 : Simplifying Inequalities

Write and simplify the inequality:   Two times the quantity of two less than a number squared is less than four.

Explanation:

Write the inequality in parts before simplifying.

A number squared:

Two less than a number squared:

Two times the quantity of two less than a number squared:

Is less than four:

Divide by two on both sides.

Simplify both sides.

Add two on both sides.

Simplify both sides.

Square root both sides.

This will split into two solutions.

The first answer is:

There will be a negative component as well for the second answer.

Dividing by a negative one on both sides will result in switching the sign.

Example Question #8 : Simplifying Inequalities

Consider the following equality:

Which of the following gives the same solution set?

Explanation:

Can be written as a compound inequality statement:

or

Solving these inequalities gives:

Example Question #9 : Simplifying Inequalities

Consider the following inequality:

Which of the following statements has the same solution set?

or

and

or

and

or

Explanation:

1. Rewrite as a compound inequality statement:

or

2. Simplify:

becomes

becomes