Solving Compound Inequalities
As it is already known that compound inequalities are formed when we merge two or more inequalities while solving compound inequalities firstly we solve the individual inequalities and then the union or intersection of the answers to those inequalities gives the solution of the compound inequalities which depend on the condition used, i.e. AND or OR conditions.
For solving the compound inequality follow the steps discussed below,
Step 1: Write the two inequalities individually from the compound inequalities.
Step 2: Solve each of the following inequality individually using the normal methods of solving inequalities.
Step 3: Mark the solution of both inequalities on the number line.
Step 4: If the compound condition is AND take the intersection of the solution and if the compounding is OR take the union of both the solution to get the desired solution of the inequalities.
Example of Solving Compound Inequality with AND
Example: Solve the compound inequality 2 < 2x + 1 < 8
Solution:
Step 1: Mark the individual inequality from 2 < 2x + 1 < 8
2x + 1 > 2, 2x + 1 < 8
Step 2: Simplify both the inequality individually.
2x + 1 > 2
⇒ 2x > 2-1
⇒ 2x > 1
⇒ x > 1/2
AND, 2x + 1 < 8
⇒ 2x < 8 – 1
⇒ 2x < 7
⇒ x < 7/2
Step 3: Mark both solutions in the number line.
x ∈ (1/2, ∞)
x ∈ (-∞, 7/2)
Step 4: The compound condition is AND so we find the intersection of both the solution
x ∈ (1/2, ∞) ∩ (-∞, 7/2)
⇒ x ∈ (1/2, 7/2)
Example of Solving Compound Inequality with OR
Example: Solve the compound inequality 2x + 1 > 8 OR x + 3 < 23
Solution:
Step 1: Mark the individual inequality from 2x + 1 > 8 OR x + 3 < 23
2x + 1 > 8, x + 3 < 23
Step 2: Simplify both the inequality individually.
2x + 1 > 8
⇒ 2x > 8-1
⇒ 2x > 7
⇒ x > 7/2
OR, x + 3 < 23
⇒ x < 23 – 3
⇒ x < 20
Step 3: Mark both solutions in the number line.
x ∈ (7/2, ∞)
x ∈ (-∞, 20)
Step 4: The compound condition is OR so we find the intersection of both the solution
x ∈ (7/2, ∞) U (-∞, 20)
Read More,
Compound Inequalities
Compound Inequalities are the combination of two or more inequalities. These inequalities are combined using two conditions that are AND, and OR. These conditions have specific meanings and they are solved differently. The inequities in compound inequalities are individually solved using normal rules but the combinations of their answers depend on the AND and OR conditions. So, let’s start learning about the concept of compound inequalities including their solutions and various other solved examples as well.