This document discusses solving inequalities. It explains that inequalities are solved similarly to equations, with the additional rule that the inequality symbol must be reversed if multiplying or dividing by a negative number. Graphs of inequalities use open or closed circles and arrows to indicate whether values are included or excluded. The document also covers compound inequalities connected by "and" or "or", explaining that "and" means both statements must be true while "or" means either statement can be true alone. Examples are provided to demonstrate solving single and compound inequalities algebraically and graphing the solutions.

1. ≤ , ≥ , <, >
Inequalities
Chapter 6

2. Solving Inequalities
 To solve an inequality, use inverse operations and
solve it just like an equation.
 Extra Rule: If you multiply or divide both sides by
a negative when doing inverse operations, you
must reverse the inequality symbol.
 The graph is an open circle and arrow if only less
than or greater than, or a closed circle and arrow
if also equal to.

3. Example #1
 Solve: −2 x − 7 ≤ 5
+7 +7
−2 x ≤ 12
−2 -2
x ≥ −6
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Greater than or equal to means closed circle and arrow.

4. Example #2
x
 Solve:
+ 1 > −1
−4
−1 −1
x
−4 ( ) > (−2 ) ( −4 )
−4
x<8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Less than means open circle and arrow.

5. Example #3
Solve:
4x − 2 > 2x + 6

− 2x − 2x
2x − 2 > 6
+2 +2
2x > 8
x>4
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Less than means open circle and arrow.

6. Compound Inequalities
 A compound inequality is two inequalities
with the word “and” or “or” between them.
 “And” statements mean that both statements
must be true and the graph is the
intersection of the two answers.
 “Or” statements mean that only one has to
be true and the graph is the union of the two
answers.

7. Steps for Solving
Compound Inequalities
Step 1: Separate the problem as two
inequalities.
Step 2: Solve for each problem.
Step 3: Graph each answer on a single
number line. The combined
graph and inequality is your
answer.

8. Compound Example of “And”
 Solve: 3 ≤ 2 x − 3 < 15
3 ≤ 2x − 3 2 x − 3 < 15
6 ≤ 2x 2 x < 18
3≤ x x<9
3≤ x <9
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
The answer is the intersection between the two points.

9. Compound Example of “Or”
 Solve: h − 10 < −14 or h + 3 ≥ 2
h − 10 < −14 h+3≥ 2
h < −4 h ≥ −1
h < −4 or h ≥ −1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
The answer is the union of the two answers. It can be either but
not both at the same time.

10. Compound Example of “Or”
 Solve: h − 10 < −14 or h + 3 ≥ 2
h − 10 < −14 h+3≥ 2
h < −4 h ≥ −1
h < −4 or h ≥ −1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
The answer is the union of the two answers. It can be either but
not both at the same time.