Compound Inequality

Compound inequality or combined inequality includes two or more inequalities joined together either by Conjunction – â€˜ANDâ€™ or Disjunction – â€˜OR.â€™

Here are some examples of compound inequalities:

• x < 3 or x > 7
• x > 8 and x < 12
• -5 â‰¤ x â‰¤ 10

Types

The two types of Compound inequalities are:

Conjunction – â€˜ANDâ€™

In such inequalities, two conditions are true at the same time.

For example, x > 8 and x < 12 and -5 â‰¤ x â‰¤ 10 are examples of â€˜ANDâ€™ – type compound inequality. -5 â‰¤ x â‰¤ 10 means the value of x is greater than equal to -5 and, simultaneously, less than equal to 10.

Disjunction – â€˜ORâ€™

In these inequalities, one of the two conditions is true.

For example, in the â€˜ORâ€™ – type compound inequality x < 3 or x > 7, the value of x is either less than 3 or more than 7. Both of them cannot be true.

Solving

Solving a compound inequality follows the same steps as solving a linear inequality. However, there are differences based on the inequality, whether it is a conjunction or a disjunction.

Conjunctions

Let us solve the inequality -4 < 2x + 6 < 4

Subtracting 6 from both sides, we get

-4 – 6 < 2x + 6 – 6 < 4 – 6

â‡’ -10 < 2x < -2

On dividing both sides by 2, we get

-5 < x < -1 (Also, x > -5 and x < -1)

Thus, solving conjunctions involves solving each inequality separately, and the overlapping section is the solution of the given inequality.

Disjunctions

Similarly,  let us solve the inequality x – 5 < -4 or x – 5 > 4

Adding 5 on both sides, we get

x – 5 + 5 < -4 + 5 or x – 5 + 5 > 4 + 5

â‡’ x < 1 or x > 9

Thus, solving disjunctions involves solving each inequality separately and then uniting the solutions.

Graphing

Graphing a compound inequality follows the same steps as graphing a linear inequality. However, there are differences depending on whether the inequality is a conjunction or a disjunction.

Conjunctions

Let us graph the inequality -5 < x < -1 (x > -5 and x < -1).

We observe that the solutions of inequalities are graphed on the number line as rays, where the open circle denotes x â‰  -5 and x â‰  -1.

They are also expressed using the â€˜intervalâ€™ notation or the â€˜intersectionâ€™ symbol (âˆ©) between the intervals.

Here, the solution is (-5, -1) or (-5, âˆž) âˆ© (-âˆž, -1).

Now, plotting the graph of another compound inequality -5 â‰¤ x â‰¤ 10 (x â‰¥ -5 and x â‰¤ 10), we get

Here, the closed circle denotes that â€˜xâ€™ can equal -5 or â€˜xâ€™ can equal 10.

Thus, [-5, 10] or [-5, âˆž) âˆ© (-âˆž, 10] is the required solution.

Disjunctions

Now, let us graph the compound inequality x < 1 or x > 9

Graphing each inequality on the number line, we get

In the graph, the open circle denotes x â‰  1 and x â‰  9. Using the â€˜unionâ€™ symbol (âˆª) between the intervals, the above compound inequality can be written as (-âˆž, 1) âˆª (9, âˆž)

Thus, the solution is (-âˆž, 1) âˆª (9, âˆž)

Now, let us graph another inequality x < 3 or x > 7

Here, the open circle denotes x â‰  3 and x â‰  7, and thus, the required solution is (-âˆž, 3) âˆª (7, âˆž)

Solved Examples

Solve the compound inequalities 2x – 5 > -11 and 2x + 1 < 9 and write the solution in the interval notation.

Solution:

Here the given inequalities are 2x – 5 > -11 and 2x + 1 < 9
On solving, we get
â‡’ 2x – 5 + 5 > -11 + 5 (by addition property) and 2x + 1 – 1 < 9 – 1 (by subtraction property)
â‡’ 2x > -6 and 2x < 8
â‡’ x > -3 (by division property) and x < 4 (by division property)
Now, graphing the inequalities separately,Â
Thus, the solution is (-3, 4)

Solve the compound inequalities 5x – 4 â‰¤ 1 or 8x – 8 â‰¥ -8 and find the solution.

Solution:

Here the given inequalities are 5x – 4 â‰¤ 1 or 8x – 8 â‰¥ -8
â‡’ 5x – 4 + 4 â‰¤ 1 + 4 (by addition property) or 8x – 8 + 8 â‰¥ -8 + 8
â‡’ 5x â‰¤ 5 or 8x â‰¥ 0
â‡’ x â‰¤ 1 (by division property) or x â‰¥ 0 (by division property)
Now, graphing the inequalities separately,
Thus, the solution is the set of all real numbers.

Which compound inequality is represented by the following graph?

Solution:

Here, the given graph represents the compound inequality -4 â‰¤ x < 1 for the variable x.

About 10% of John’s time per day is spent watching TV. How long does he spend in about 2 to 3 days? Write and graph the compound inequality of the situation.

Solution:

Here, 10% of 1 day (24 hours) = 2.4 hours
Using the interval notation, we conclude that John spends (4.8, 7.2) hours in 2 to 3 days, and its graph is plotted.