1. Summer Assignment Review
Graphing Compound Inequalities
And Absolute Value Inequalities

2. Compound Inequalities
AND compound inequalities should create a range of values.
−2≤𝑥≤3
𝑥≥−2 𝐴𝑁𝐷 𝑥≤3
Don’t assume that the given inequality creates this range.
−2≥𝑥≥3
𝑥≤−2 𝐴𝑁𝐷 𝑥≥3
𝑥 falls between -2 and 3 on the number line.
𝑥 can not be both to the left of -2 and to the right of 3 on the number line . There is no solution.

3. Compound Inequalities
OR compound inequalities should create two ranges of values.
𝑥≤−2 𝑂𝑅 𝑥≥3
AND compound inequalities should create a range of values.
𝑥≥−2 𝑂𝑅 𝑥≤3
𝑥 is either to the left of -2 or to the right of 3 on the number line.
𝑥 can be anywhere on the number line. The solution is All Real Numbers.

4. Practice Graph each compound inequality. 1) −3<𝑥<4 2) 3)4) 5)
−3<𝑥<4
All Real Numbers
𝑥 ≥−3 𝑜𝑟 𝑥<3
𝑥<−4 𝑜𝑟 𝑥 ≥2
𝑥≤0 𝑎𝑛𝑑 𝑥≥3
No Solutions
𝑥>−1 𝑎𝑛𝑑 𝑥 ≤5

5. Absolute Value Inequalities
When we say that 𝑥 =3, we are saying that the distance between x and 0 is 3.
When we say that 𝑥 <3, we are saying that the distance between x and 0 is less than 3.
When we say that 𝑥 >3, we are saying that the distance between x and 0 is greater than 3.
The inequalities < and ≤ lead to “and” relationships
−3<𝑥<3
The inequalities > and ≥ lead to “or” relationships
𝑥<−3 𝑂𝑅 𝑥>3

6. Solving Absolute Value Inequalities
Case 2 - modified method: Drop the absolute value bars, flip the inequality symbol and change the sign of the term on the right.
2𝑥−1<−7
2𝑥<−6
𝑥<−3

7. Solve and Graph an Abs. Val. Ineq.
Example : Solve 𝑥−5 ≥7 . Graph your solution.
Step 1: Is it and or or?
For the second, flip the inequality and
change the sign of the 7
Step 2: Solve both
Step 3: Graph
This is ≥ so it is “or”. Set up two inequalities.
𝑥−5 ≥7
𝑥−5≥7
𝑥−5≤−7
𝑥≥12
𝑥≤−2
𝑥≥ or
𝑥≤−2
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8. Solve and Graph an Abs. Val. Ineq.
Example : Solve Graph your solution.
Step 1: Is it and or or?
Step 2: Solve
Step 3: Graph
−4𝑥−5 +3<9
We don’t know yet, get the abs. val. alone.
Now :
It is < , so it is an “and” inequality. Drop the abs. val. Bars and put -6 < on the left.
−4𝑥−5 +3<9
−4𝑥−5 <6
−6<−4𝑥−5<6
−1<−4𝑥<11
0.25>𝑥>−2.75
−2.75<𝑥<0.25

9. Practice −5<𝑤<6 −0.2≤𝑚≤2.6 𝑥>5 𝑜𝑟 𝑥<−11
𝑥>5 𝑜𝑟 𝑥<−11
−5<𝑤<6
−0.2≤𝑚≤2.6
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