Objective- To solve compound inequalities involving

1
Aim How do we solve Compound Inequalities?
Do Now
Solve the following inequalities
1. 2x 3 gt 2
2. 5x lt 10
How do we put two inequalities together?
2
Definition A compound inequality consists of two
inequalities connected by the word and or the
word or.
Examples -7 lt x lt 10 x lt 8 or x gt 27 x lt -
4 or x gt 4 12 ? x and x ? 30
3

• Example
• This is a conjunction because the two inequality
  statements are joined by the word and.
• You must solve each part of the inequality.
• The graph of the solution of the conjunction is
  the intersection of the two inequalities. Both
  conditions of the inequalities must be met.
• In other words, the solution is wherever the two
  inequalities overlap.
• If the solution does not overlap, there is no
  solution.

4
and Statements can be Written in Two Different
Ways

• 1. 8 lt m 6 lt 14
• 2. 8 lt m6 and m6 lt 14
• These inequalities can be solved using two
  methods.

5
Method One

• Example 8 lt m 6 lt 14
• Rewrite the compound inequality using the
  word and, then solve each inequality.
• 8 lt m 6 and m 6 lt 14
• 2 lt m m lt 8
• m gt2 and m lt 8
• 2 lt m lt 8
• Graph the solution

6
Method Two

• Example 8 lt m 6 lt 14
• To solve the inequality, isolate the variable by
  subtracting 6 from all 3 parts.
• 8 lt m 6 lt 14
• -6 -6 -6
• 2 lt m lt 8
• Graph the solution.

7
Review of the Steps to Solve a Compound
Inequality

• Example
• This is a disjunction because the two inequality
  statements are joined by the word or.
• You must solve each part of the inequality.
• The graph of the solution of the disjunction is
  the union of the two inequalities. Only one
  condition of the inequality must be met.
• In other words, the solution will include each of
  the graphed lines. The graphs can go in opposite
  directions or towards each other, thus
  overlapping.
• If the inequalities do overlap, the solution is
  all reals.

8
or Statements

• Example x - 1 gt 2 or x 3 lt -1
• x gt 3 x lt -4
• x lt -4 or x gt3
• Graph the solution.

9
Solve and graph the compound inequality.
and
and
-7 0 4
10
Solve and graph.
-3 0 1
11
Solve and graph.
-2 0 10
12
Solve and graph the compound inequality.
or
or
or
0 5 10
13
Solve and graph the compound inequality.
or
or
-7 0 12
14
Number Line Graphs of Inequalities
Intersections
Unions
x lt 5 x lt 3
x lt 5 x lt 3
0 1 2 3 4 5 6
0 1 2 3 4 5 6
x x lt 3
x x lt 5
x lt 5 x gt 3
x lt 5 x gt 3
0 1 2 3 4 5 6
0 1 2 3 4 5 6
x 3 lt x lt 5
x x Any Real Number
15
Number Line Graphs of Inequalities
Intersections
Unions
x gt 5 x lt 3
x gt 5 x lt 3
0 1 2 3 4 5 6
0 1 2 3 4 5 6

x x lt 3 or x gt 5
x gt 5 x gt 3
x gt 5 x gt 3
0 1 2 3 4 5 6
0 1 2 3 4 5 6
x x gt 5
x x gt 3