Compound Inequality

Compound inequality or combined inequality includes two or more inequalities joined together either by Conjunction – ‘AND’ or Disjunction – ‘OR.’

Here are some examples of compound inequalities:

• x < 3 or x > 7
• x > 8 and x < 12
• -5 ≤ x ≤ 10

Types 

The two types of Compound inequalities are:

Conjunction – ‘AND’

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

For example, x > 8 and x < 12 and -5 ≤ x ≤ 10 are examples of ‘AND’ – type compound inequality. -5 ≤ x ≤ 10 means the value of x is greater than equal to -5 and, simultaneously, less than equal to 10. 

Disjunction – ‘OR’

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

For example, in the ‘OR’ – 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

⇒ -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

⇒ 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). 

Figure 1 (Heading: compound Inequalities, Filename: Solve the Compound Inequalities)

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

They are also expressed using the ‘interval’ notation or the ‘intersection’ symbol (∩) between the intervals. 

Here, the solution is (-5, -1) or (-5, ∞) ∩ (-∞, -1).

Now, plotting the graph of another compound inequality -5 ≤ x ≤ 10 (x ≥ -5 and x ≤ 10), we get

Figure 2 (Heading: compound Inequalities, Filename: Graph the Compound Inequalities) 

Here, the closed circle denotes that ‘x’ can equal -5 or ‘x’ can equal 10.

Thus, [-5, 10] or [-5, ∞) ∩ (-∞, 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 

Figure 3 (Heading: compound Inequalities, Filename: Solving Compound Inequalities)

In the graph, the open circle denotes x ≠ 1 and x ≠ 9. Using the ‘union’ symbol (∪) between the intervals, the above compound inequality can be written as (-∞, 1) ∪ (9, ∞)

Thus, the solution is (-∞, 1) ∪ (9, ∞)

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

Figure 4 (Heading: compound Inequalities, Filename: Graphing Compound Inequalities) 

Here, the open circle denotes x ≠ 3 and x ≠ 7, and thus, the required solution is (-∞, 3) ∪ (7, ∞)

Solved Examples