Problem 115 - College Algebra

Question

What is a compound inequality and how is it solved?

Step-by-Step Solution

Answer

A compound inequality contains at least two inequalities that are separated by either 'and' or 'or'. The ones that are separated by 'and' are often occurring between the two stated inequalities, whereas the ones separated by 'or' are normally not between the stated inequalities. For instance, for an 'and' inequality -2 < x - 1 < 5, it can be solved as -1 < x < 6. For an 'or' inequality x + 3 < -2 or x - 2 > 5, it can be solved as x < -5 or x > 7.

Step 1: Understanding the concept

A compound inequality consists of two or more inequalities joined together with either the word 'and' or the word 'or'. For instance, a compound inequality with the word 'and' could be written as -2 < x < 5, which means x is greater than -2 and x is less than 5. On the other hand, a compound inequality that uses the word 'or' could be written as x < -2 or x > 5, which means x is less than -2 OR x is greater than 5.

Step 2: Solving Compound Inequalities of the type 'AND'

Let's take an example -2 < x - 1 < 5. Here, to find the constitutional elements of x, the first thing to do is to isolate x. This could be done by adding 1 throughout the inequality so as to get x alone. This would give -2 + 1 < x < 5 + 1; which simplifies to -1 < x < 6. This means that x is greater than -1 and x is less than 6.

Step 3: Solving Compound Inequalities of the type 'OR'

As for an 'OR' type of inequality, let's consider an example x + 3 < -2 or x - 2 > 5. The inequalities can be solved separately. Therefore, for the first term, by subtracting 3 everywhere, we have x < -2 - 3, which simplifies to x < -5 and for the second term, which is x - 2 > 5, by adding 2 to all sides, we have x > 5 + 2, which simplifies to x > 7. This means that x is less than -5 OR x is greater than 7.