# Solutions to Problem Set 5

Math 101-06

9-13-2017

[Absolute value equations and inequalities]

1. Solve for x:

2. Solve for x:

An absolute value can't be negative. The equation has no solutions.

3. Solve for x:

4. Solve for x. Write your answer using either interval notation or inequality notation.

Solve the corresponding equation:

I put -3 and 5 on the number line. Since the absolute value expression is on the greater-than ("open") side of the " ", I shade the outside intervals:

The solution is or .

In interval notation, this is .

5. Solve for x. Write your answer using either interval notation or inequality notation.

First, solve the helper equation .

I put -12 and 8 on the number line. Since the absolute value expression is on the less-than ("pointy") side of the " ", I shade the inside interval:

The solution is .

In interval notation, this is .

6. Solve for x. Write your answer using either interval notation or inequality notation.

An absolute value can't be negative. Hence, there are no solutions.

7. Solve for x. Write your answer using either interval notation or inequality notation.

An absolute value is always greater than or equal to 0. So is always greater than a negative number like -5, no matter what x is. Hence, the solution is all real numbers (or , or ).

8. Solve for x. Write your answer using either interval notation or inequality notation.

Solve the corresponding equation:

I put -1 and 6 on the number line. Since the absolute value expression is on the less-than ("pointy") side of the " ", I shade the inside interval:

The solution is .

In interval notation, this is .

9. Solve for x. Write your answer using either interval notation or inequality notation.

Solve the corresponding equation:

I put -11 and 31 on the number line. Since the absolute value expression is on the less-than ("pointy") side of the " ", I shade the inside interval:

The solution is .

In interval notation, this is .

We are what we think. All that we are arises with our thoughts. With our thoughts we make the world. - The Buddha

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