Problem 138 - College Algebra

Question

Describe how to solve an absolute value inequality involving the symbol <. Give an example.

Step-by-Step Solution

Answer

For absolute value inequality |ax + b| < c, split the inequality into two: ax + b < c and -ax - b < c. Solve each one separately to get two solution intervals, and take the intersection (common part) of these intervals as the final solution. For example, the solution to the inequality |2x + 3| < 7 is the interval (-5,2).

Step 1: Isolate the absolute value

Given an inequality of the form |ax + b| < c, where a, b and c are real numbers and x is the variable to solve for, the first task is to isolate the absolute value. If necessary, this might involve operations such as adding, subtracting, multiplying or dividing both sides of the inequality.

Step 2: Split the inequality

Once the absolute value is isolated, split the inequality into two separate inequalities based on the definition of absolute value: ax + b < c and -ax - b < c. Keep in mind to switch the direction of the inequality when multiplying or dividing by a negative number.

Step 3: Solve the inequalities

Subsequently, solve each of the inequalities for x independently. This might involve further simplification such as adding or subtracting terms, or multiplying or dividing by coefficients.

Step 4: Find the Intersection of Solutions

Since the original inequality involves a 'less than' absolute value, the solution is the intersection (common part) of the solutions to both inequalities obtained in Step 3. Write the solution in interval notation.

Step 5: Example

For instance, to solve |2x + 3| < 7, in step 1 the absolute value is already isolated. Apply the second step to get the two inequalities: 2x + 3 < 7 and -2x - 3 < 7. Solving these two inequalities gives x < 2 and x > -5, respectively. The common part of the solutions is -5 < x < 2, which is the interval (-5,2).