Solving Compound Inequality

Two inequality statements joined either by  conjunction or disjunction is known as compound inequality. Conjunction shown by the word AND and disjunction is denoted by Or.    “And” shows that both statements of the compound sentence are true at the same time. It is the intersection of the two setst.  “Or” denotes that, as long as either statement is true, the entire compound sentence is true. It is the combination or union of the answer sets .

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Ex :           2x-3<5 and="" x="">13

5n>65 or n+7<3 p="">
Solving Compound inequality Example problems:

Ex 1 :   Solve the inequality: 2x-3<7 and="" x="">12

Sol :     Step 1 : Solve the first inequality

2x-3<7 3="" both="" dd="" on="" p="" sides="">
2x<10 2="" both="" by="" divide="" on="" ow="" p="" sides="">
x<5 p="">
Step 2:  solve the second inequali

x+14>12 (Subtract 14 from both sides

x>-2

By solving compound inequality we get the final solution is:

x>-2 and x<5 p="">
This means that all numbers between -2 and 5 are solutions

Ex 2:  Solve the compound inequality 4x + 5 < 17     or     4x > 28

Sol :  Step 1 : Solve the first inequality

4x+5 < 17

Subtract 5 on both sides

4x+5-5 <17-5 p="">
4x<12 p="">
Divide by 4 on both sides,

x<3 p="">
Step 2: Solve the second  inequality

4x>28

Divide by 4 on both sides

x>7

By solving compound inequality the final solution as ,

7
Solving Compound double inequality problems:

Ex 1 :  Solve the following double inequality

-3 < 2x + 5 < 7

Sol :  This can be written as a compound inequality by writing

-3  <  2x + 5  and  2x + 5  <  7

Solving compound inequality,

Subtract 5 on both sides for both inequalities

-8  <  2x  and  2x  <  2

Divide by 2 on both sides for both inequalities

-4  <  x  and  x  <  1

Ex 2:     Solve    -5 < 3x + 4 < 19 .

Sol :       -5  <  3x + 4  <  19

Step 1:  Write the double inequality as compound inequality

-5<3x 3x="" and="" p="">
Step 2: Solving compound inequality,

Subtract 4 on both sides for both inequalities

-5-4<3x 3x="" and="" nbsp="" p="">
-9<3x 3x="" and="" nbsp="" p="">
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Step 3:

Divide by 3 on both sides for both inequalitie

-3
The final solution is,

x>-3 and x<2 p="">