**Inequalities, Absolute Value, Solving, Graphing and Real-World Examples.**

Inequalities

An equation where you use the less than or greater than sign to compare numbers. ex: a<b

An inequality is a statement in which two expressions are

not

equal.

Examples:

a < b

a > b < c

a ≤ b

a ≥ b > c

Solving Inequalities

When solving an inequality there should be one variable, a constant, an inequality sign and another constant.

To solve you have to subtract or add the constant in front of the inequality to the one after the sign.

Examples:

x + 9 > 10

- 9 - 9

x > 1

Graphing Inequalities

There are two steps to graphing inequalities.

Step 1: Split the inequality into two parts

Step 2: Graph each inequality

Also, you have to remember that if you have an inequality like < or > than the circle is an open one unlike the closed one that occurs when the sign is ≥ or ≤

Examples:

4<x<10

4<x and x<10

4 10

Real-World Inequality Examples

Parts for an automobile repair cost $175. The mechanic charges $34 per hour. If you receive an estimate for at least $226 and at most $268.50 for fixing the car, what is the time interval that the mechanic will be working the job?

7x - 3 ≤ 9 + 3x

A museum hires students for part-time work at $5.25 an hour. Each student works from 12 to 26 hours per week. Finde the range of weekly salaries for the students.

3 < x - 3 ≤ 6

Absolute Value

Absolute Value means how far the number is from zero.

When doing absolute value you have to consider that the quantity inside is positive and the quantity inside could also be negative.

Example: |d - 3| = 7 |d - 3| = -7

+ 3 + 3 + 3 + 3

d = 10 d = -4

**By: Gianna Bencivengo and Makenzie Laidlow**

p - 23 < 48

+ 23 +23

p < 71

Solving Absolute Value

When solving absolute value, you have to make sure that you subtract or add the extra constant in the equation.

For example:

|4x + 2| -3 = 10

+3 + 3

4x + 2 = 13

And then you proceed to solve the equation like a normal one.

Graphing Absolute Value

When graphing absolute value, the points should always form a V.

When you add or subtract numbers outside the absolute value lines than it goes up or down on the y axis.

When you add or subtract numbers inside the absolute value lines than it goes right or left on the x axis.

For example:

y = |x| - 2 y = |x + 3|

Real-World Absolute Value Examples

y = |x - 8|

y = |x| +2

|4x + 2| +2 = 32

The employees were looking at their debts. Gomez is in $143.00 of debt; Cole has $203.00 of debt. Ford has $178.00 of debt. Simpson has a positive bank balance of $350.00. Who owes the most money?