Section 2.7
Absolute Value Equations & Inequalities
Big Things to Remember!!!

If you multiply or divide both sides by a
negative number, FLIP the inequality
sign.

DOUBLE CHECK to make sure you
appropriately used parentheses and
brackets.
Absolute Value
Definition of Absolute Value of a Number
Geometrically, the absolute value of a number 𝑎, written as
𝑎 , is the distance on the number line from 0 to 𝑎.

Examples:
◦ −5 = 5
◦ 𝑥 =4
Absolute Value Inequalities
So, what does something like this mean?
𝑥 ≤3
Absolute Value Inequalities
So, what does something like this mean?
𝑥 ≥3
Critical Info
When Solving Abs.Value Equations and Inequalities (with the
Absolute Value Sign on the Left) . . .
Symbols
Words Examples
>
First Step in Solving
𝑥 + 1 > 12
𝑥 + 1 > 12 or 𝑥 + 1 < −12
2𝑥 ≥ 4
2𝑥 ≥ 4 or 2𝑥 ≤ −4
=
2 − 5𝑥 = 20
2 − 5𝑥 = 20 or 2 − 5𝑥 = −20
<
𝑥 + 1 < 12
𝑥 + 1 < 12 and 𝑥 + 1 > −12
2𝑥 ≤ 4
2𝑥 ≤ 4 and 2𝑥 ≥ −4
≥
OR
AND
≤
Example 1
Solve the equations. Write the solution set.
a.
2
𝑥
3
b.
5 − 𝑥 = −2
c.
d.
−1 =5
0 = 2𝑥 + 7
13𝑥 = 2𝑥 + 1
Example 2
Solve the inequality and graph the solution set.
−2𝑥 − 6 ≤ 5
Example 3
Solve the inequality and graph the solution set.
5−𝑥 >3
Example 4
Solve the inequality and graph the solution set.
10 ≥ 4 + −2𝑥
Example 5
Solve the inequality and graph the solution set.
a.
18 − 100,000,000,000𝑥 > −2
b.
18 − 100,000,000,000𝑥 < −2
Testers
1.
?
≥0
2.
?
> −7
3.
?
<0
4.
?
< −4
Section 3.1
The Rectangular Coordinate System
The Cartesian Plane
Example 1
Plot the ordered pairs and give the quadrant.
a.
(2, 6)
b.
(0, 0)
c.
(-3, 4)
d.
(4, 0)
e.
(0,-7)
Linear Equations
Standard Form of Linear Equation in Two Variables
𝐴𝑥 + 𝐵𝑦 = 𝐶
where 𝐴, 𝐵, and 𝐶 are real numbers with 𝐴, 𝐵 ≠ 0.

To find the intercepts . . .
◦ 𝑥-int: plug in 0 for 𝑦
and solve for 𝑥
◦ 𝑦-int: plug in 0 for 𝑥
and solve for 𝑦
Example 2
Find the intercepts. Graph the line.
3𝑥 − 7𝑦 = 9
Example 3
Find the intercepts. Graph the line.
−2𝑥 + 4𝑦 = 12
Example 4
Find the intercepts. Graph the line.
𝑦+4=0
Example 5
Find the intercepts. Graph the line.
𝑥=7
Example 6
Complete the table and graph the equation.
6𝑥 − 5𝑦 = 30
𝑥
𝑦
0
0
3
-2
Midpoint of a Line Segment
Midpoint
Formula
Example 7
Find the midpoint of the segment with
these endpoints.
4, −3 and (−1,3)
Questions???

Don’t forget to check on your due dates!