Title: Objective- To solve compound inequalities involving
1Aim 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?
2Definition 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.
4and 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.
5Method 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
6Method 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.
7Review 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.
8or 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.
9Solve and graph the compound inequality.
and
and
-7 0 4
10Solve and graph.
-3 0 1
11Solve and graph.
-2 0 10
12Solve and graph the compound inequality.
or
or
or
0 5 10
13Solve and graph the compound inequality.
or
or
-7 0 12
14Number 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
15Number 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