1. Drill
Write the compound inequality for the following graphs.

2. Drill Solve each compound inequality and graph its solution.
6 + 7m < 6m – 5 OR 3m – 7 < 5 + 6m

3. Solving Absolute Value Equations
Algebra 1 ~ Chapter 6.5
Solving Absolute Value Equations

4. * Recall that the absolute-value of a number is that number’s distance from zero on a number line. For example, |–5| = 5 and |5| = 5.
5 units
5 units 6 5 4 3 2 1 1 2 3 4 5 6
* For any nonzero absolute value, there are exactly two numbers with that absolute value. For example, both 5 and –5 have an absolute value of 5.
* To write this statement using algebra, you would write | x | = 5. This equation asks, “What values of x have an absolute value of 5?” The solutions are 5 and –5. Notice this equation has two solutions.

5. To solve absolute-value equations, perform inverse operations to isolate the absolute-value expression on one side of the equation. Then you must consider two cases.

6. Ex. 1 - Solve the equation. The solution is x = 12, –12. |x| = 12
Think: What numbers are 12 units from 0?
12 units
10 8 6 4 2 4 6 8 10
12 2 12 •
Case 1
x = 12
Case 2
x = –12
The solution is x = 12, –12.

7. Ex. 2 - Solve the equation. Think: What numbers are 4 units from 0?
CHECK both solutions:
The solution is x =-3, –11

8. Ex. 3 - Solve the equation. Think: What numbers are 7 units from 0?
CHECK both solutions:
The solution is x = -1, 6

9. Ex. 4 - Solve the equation. Think: What numbers are 1 unit from 0?
CHECK both solutions:
2 – ⅓x = -1
– ⅓x = -3
x = 9
2 – ⅓x = 1
– ⅓x = -1
x = 3
The solution is x = 3, 9

10. Writing Absolute Value Equations given the solution:6 5 4 3 2 1 1 2 3 4 5 6
Steps:
Find the midpoint of the two values (x1 +x2)
This is what you subtract from x:
2) Calculate the distance from that point to each solution
This goes on the right side of the =
3) Write the equation.

11. Example: You try it… Steps: Find the midpoint6 5 4 3 2 1 1 2 3 4 5 6
Steps:
Find the midpoint
2) Calculate the distance from that point to each solution
3) Write the equation.

12. Example: You try it… 6 5 4 3 2 1 1 2 3 4 5 6