Inequality is the relation between two expressions that are not equal, and having a sign such as $\neq$ (not equal to), $>$ (greater than), $<$ (less than), $\geq$ (greater than or equal to), or $\leq$ (less than or equal to).  A compound inequality contains more than one inequality that is it is a combination of two or more inequalities. These inequalities can be joined together with either OR or AND operations. The solution set of a compound inequality with "and" in between is the set of all elements common to the solution sets of both inequalities. The solution set of a compound inequality with "or" in between is the union of the solution sets of the two inequalities. If the same number is added to or subtracted from both sides of an inequality, the resulting inequality has the same solution as the original inequality, that is there will be no change in the solution set. If both sides of an inequality are multiplied or divided by the same positive number, the resulting inequality has the same solution as the original inequality, that is there will be no change in the solution set. If both sides of an inequality are multiplied or divided by the same negative number, then we reverse inequality sign to get the solution set.

## Definition

A compound inequality (or combined inequality ) is two or more inequalities joined together. Inequalities can be joined either with OR or AND. To be a solution of an OR  inequality, a value has to make only one part of the inequality true, it means if we take any value from the solution set, it must satisfy either one of the inequality. To be a solution of an AND inequality, it must make both parts true, it means if we take any value from the solution set, it must satisfy both the inequalities. To solve a compound inequality, first we need to separate it into two inequalities and then solve each inequality separately. Determine whether the answer should be a union of sets ("OR") or an intersection of sets ("AND") and then write the solution set carefully and then graph it on the number line. If there is a doubt about whether the inequality is a union of sets or an intersection of sets, then we can take one value from each region and then test each region to see if it satisfies the compound inequality. If the compound inequality is OR then it must satisfy any one inequality. If the compound inequality is AND then it must satisfy both the inequalities.

## Compound Inequality OR

A compound inequality is a combination of two or more inequalities. A compound inequality OR is a combination of two or more inequalities with a OR in between. To be a solution of an OR  inequality, a value has to make only one part of the inequality true, it means if we take any value from the solution set, it must satisfy either one of the inequality. To solve a compound inequality, first we need to separate it into two inequalities and then solve each inequality separately. Here the compound inequality is OR so the value taken from the solution set must satisfy any one inequality.
Example:

Solve the compound inequality $2x + 3 \leq 17$ or $x + 5 \geq 9$?

Solution:

Step 1:

Solve each inequality separately

First inequality is

$2x + 3 \leq 17$

Subtracting $3$ on both sides, we get

$2x + 3 - 3 \leq 17 - 3$

Simplification

$2x \leq 14$

Dividing both sides by $2$

$\frac{2x}{2}$ $\leq$ $\frac{14}{2}$

Simplification

$x \leq 7$

Graphing first inequality on number line

Solution set of first inequality is $\{ x|x \leq 7 \}$

Second inequality is

$x + 5 \geq 9$

Subtracting $5$ on both sides, we get

$x + 5 - 5 \geq 9 - 5$

Simplification

$x \geq 4$

Graphing second inequality on number line.

Solution set of second inequality is $\{ x|x \geq 4 \}$

Step 2:

As the compound inequality is OR the solution set will be the union of both sets

Solution set is $\{ x|x \leq 7$ or $x \geq 4 \}$

Graphing compound inequality.

## Compound Inequality AND

A compound inequality AND is a combination of two or more inequalities with a AND in between. To be a solution of an And  inequality, a value has to make both parts of the inequality true, it means if we take any value from the solution set, it must satisfy both inequalities. To solve a compound inequality, first we need to separate it into two inequalities and then solve each inequality separately. Here the compound inequality is AND so the value taken from the solution set must both inequalities.
Example:

Solve the compound inequality $3x - 6 \leq 24$ and $x - 1 \geq 2$?

Solution:

Step1:

Solve each inequality separately

First inequality is

$3x - 6 \leq 24$

Adding $6$ on both sides, we get

$3x - 6 + 6 \leq 24 + 6$

Simplification

$3x \leq 30$

Dividing by three on both sides, we get

$\frac{3x}{3}$ $\leq$ $\frac{30}{3}$

Simplification

$x \leq 10$

Graphing first inequality on number line.

Solution set for first inequality is $\{ x|x \leq 10 \}$

Second inequality is

$x - 1 \geq 2$

Adding 1 on both sides, we get

$x - 1 + 1 \geq 2 + 1$

Simplification

$x \geq 3$

Graphing second inequality

Step 2:

As the compound inequality is connected by AND, we need to take all the values that are common in both solution sets.

So the solution set of the compound inequality becomes $\{ x|3 \leq x \leq 10 \}$

Graphing compound inequality

## Compound Inequality with Fractions

To solve compound inequality involving fractions, it would be easy if we get rid of the inequality having fractions by multiplying both sides of the equation by the LCM of all the denominators. Then we can follow the same procedure as we solve compound inequalities.
Example:

Solve the compound inequality  $5 <$ $\frac{2x + 6}{4}$ $< 12$

Solution:

$5 <$ $\frac{2x + 6}{4}$ $< 12$

Multiplying each term by $4$, we get

$5 \times 4 <$ $\frac{2x + 6}{4}$ $\times 4 < 12 \times 4$

Simplification

$20 < 2x + 6 < 48$

Subtracting each term by $6$ , we get

$20 - 6 < 2x + 6 - 6 < 48 - 6$

Simplification

$14 < 2x < 42$

Dividing each term by $2$, we get

$\frac{14}{2}$ $<$ $\frac{2x}{2}$ $<$ $\frac{42}{2}$

Simplification

$7 < x < 21$

Solution set $\{ x|7 < x < 21 \}$

Graphing compound inequality