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First, isolate the absolute value bar expression.

Linear Absolute Value Inequality: Solving Algebraically

Example 1: Solve .012234 x

12234 x

323 x

323323 xorx

Next, examine the constant on the right side. If it is positive or zero proceed as follows. Write two new inequalities. Form one by dropping the absolute value bars, changing the sign of the constant term and changing the direction of the inequality symbol. Form the other by dropping the absolute value bars only. Place the word "or" between them.

Table of Contents

Linear Absolute Value Inequality: Solving Algebraically

Slide 2

Now solve each inequality. The "or" means that any number that satisfies either of the inequalities will be a solution of the original inequality.

323323 xorx

5313 xorx

35

31 xorx

The solution set in interval notation is .,35

31

,

Notes: If after isolating the absolute value expression the constant term on the right is negative, the solution set of the inequality would be (- , ) because any real number substituted for x will cause the absolute value expression to produce a number greater than a negative number.

Table of Contents

Linear Absolute Value Inequality: Solving Algebraically

Slide 3

Example 2: Solve .012234 x

First, isolate the absolute value bar expression.

12234 x

323 x

323323 xandx

Next, examine the constant on the right side. If it is positive or zero proceed as follows. Write two new inequalities. Form one by dropping the absolute value bars, changing the sign of the constant term and changing the direction of the inequality symbol. Form the other by dropping the absolute value bars only. Place the word "and" between them.

Table of Contents

Linear Absolute Value Inequality: Solving Algebraically

Slide 4

Now solve each inequality. The "and" means that only those numbers that satisfy both of the inequalities will be solutions of the original inequality.

323323 xandx

5313 xandx

35

31 xandx

The solution set in interval notation is .35

,31

Notes: If after isolating the absolute value expression the constant term on the right is negative, the inequality would have no solutions because no real number substituted for x will cause the absolute value expression to produce a number less than a negative number.

Table of Contents

Linear Absolute Value Inequality: Solving Algebraically