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Last modified on May 11th, 2024

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

The two types of Compound inequalities are:

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.

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 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.

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.

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 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.

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 ≠ -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

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.

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 ≠ 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

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

**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

⇒ 2x – 5 + 5 > -11 + 5 (by addition property) and 2x + 1 – 1 < 9 – 1 (by subtraction property)

⇒ 2x > -6 and 2x < 8

⇒ x > -3 (by division property) and x < 4 (by division property)

Now, graphing the inequalities separately,

Thus, the solution is (-3, 4)

**Solve the compound inequalities 5x – 4 ≤ 1 or 8x – 8 ≥ -8 and find the solution.**

Solution:

Here the given inequalities are 5x – 4 ≤ 1 or 8x – 8 ≥ -8

⇒ 5x – 4 + 4 ≤ 1 + 4 (by addition property) or 8x – 8 + 8 ≥ -8 + 8

⇒ 5x ≤ 5 or 8x ≥ 0

⇒ x ≤ 1 (by division property) or x ≥ 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 ≤ 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.

Last modified on May 11th, 2024