SOLVING COMPOUND INEQUALITIES EXAMPLES
Compound Inequalities with "And"
A compound inequality containing and is true only if both inequalities are true. Thus, the graph of a compound inequality containing and is the intersection of the graphs of the two inequalities.
In other words, the solution must be a solution of both inequalities.
The intersection can be found by graphing each inequality and then determining where the graphs overlap.
Compound Inequalities with "Or"
Another type of compound inequality contains the word or. A compound inequality containing or is true if one or more of the inequalities is true. The graph of a compound inequality containing or is the of the graphs of the two inequalities.
In other words, the solution of the compound inequality is a solution of either inequality, not necessarily both. The union can be found by graphing each inequality.
Example 1 :
Solve the compound inequality. Then graph the solution set
k + 2 > 12 and k + 2 ≤ 18
Solution :
|
k + 2 > 12 Subtract by 2 on both sides k + 2 - 2 > 12 - 2 k > 10 |
k + 2 ≤ 18 Subtract by 2 on both sides k + 2 - 2 ≤ 18 - 2 k ≤ 16 |
By graphing the inequality k > 10, we get the graph given below.
By graphing the inequality ≤ 16, we get the graph given below.
By combining the above two graphs, we get the common region between 10 and 16.
Example 2 :
Solve the following inequality and graph the solution
3 < 2x - 3 < 15
Solution :
3 < 2x - 3 < 15
Add 3 on both sides
3 + 3 < 2x – 3 + 3 < 15 + 3
6 < 2x < 18
Divide by 2
6/2 < 2x/2 < 18/2
3 < x < 9
Example 3 :
Solve the following inequality and graph the solution
3t – 7 ≥ 5 and 2t + 6 ≤ 12
Solution :
3t – 7 ≥ 5 and 2t + 6 ≤ 12
|
3t – 7 ≥ 5 Add 7 on both sides 3t – 7 + 7 ≥ 5 + 7 3t ≥ 12 Divide by 3 on both sides 3t/3 ≥ 12/3 t ≥ 4 |
2t + 6 ≤ 12 Subtract 6 on both sides 2t + 6 – 6 ≤ 12 – 6 2t ≤ 6 Divide by 2 on both sides t ≤ 3 |
t ≥ 4 and t ≤ 3
Example 4 :
Solve the following inequality and graph the solution
4m - 5 > 7 or 4m - 5 < -9
Solution :
|
4m - 5 > 7 Add 5 on both sides 4m - 5 + 5 > 7 + 5 4m > 12 Divide by 4 on both sides 4m/4 > 12/4 m > 3 |
4m - 5 < -9 Add 5 on both sides 4m - 5 + 5 < -9 + 5 4m < -4 Divide by 4 on both sides 4m/4 < -4/4 m < -1 |
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