Limits and Continuity: Background You’ll Need 2

  • Solve inequalities that include absolute values

Solving Absolute Value Inequalities

Absolute value inequalities are pivotal in calculus for understanding limits, a foundational concept that delves into function behavior near specific points.

An absolute value inequality, such as [latex]|A| < B,|A|\le B,|A| > B,\text{or }|A|\ge B[/latex], represents conditions where [latex]A[/latex], an expression of the variable [latex]x[/latex], falls within a specific range from zero. The inequality [latex]|A| < B[/latex] is mathematically equivalent to [latex]-B< A< B[/latex].

absolute value inequality

An absolute value inequality is an equation of the form

[latex]|A| < B,|A|\le B,|A| > B,\text{or }|A|\ge B[/latex],

where [latex]A[/latex], and sometimes [latex]B[/latex], represents an algebraic expression dependent on a variable [latex]x[/latex].

Remember that absolute value is like measuring how far a number is from zero on a number line. It doesn’t matter which direction you go—left or right—the absolute value is always the distance without signs.

Solving these inequalities is about determining all the possible values for [latex]x[/latex] that meet the specified conditions, often leading to a specific interval or set of intervals.

There are two basic approaches to solving absolute value inequalities: the graphical and the algebraic approach.

The graphical method involves visually interpreting the solutions on a graph, which can give a good approximate understanding of where the solutions lie. However, the algebraic approach, though potentially more abstract, provides precise solutions that are sometimes challenging to discern graphically.

How To: Solving Absolute Value Inequalities

Algebraic Method:

  1. Isolate the absolute value expression on one side of the inequality.
  2. Set up two separate inequalities: one for the positive and one for the negative scenario.
  3. Solve both inequalities for [latex]x[/latex] and combine the solution sets.

Graphical Method:

  1. Graph the functions inside the absolute value and their opposites.
  2. Find the points of intersection with the reference value.
  3. The solution interval is between these intersection points.

Solving Inequalities Involving [latex]x[/latex]

To solve an inequality for [latex]x[/latex], follow these steps:

  1. Isolate [latex]x[/latex]: Ensure [latex]x[/latex] is by itself on one side of the inequality. If a number is added or subtracted from [latex]x[/latex], counteract this by doing the opposite operation on both sides of the inequality.
  2. Simplify: Combine like terms and simplify each side of the inequality.
  3. Divide or Multiply: If [latex]x[/latex] is multiplied by a coefficient, divide both sides of the inequality by that number to solve for [latex]x[/latex]. Remember, if you multiply or divide by a negative number, you must flip the direction of the inequality sign!
  4. Check Your Solution: Substitute your solution back into the original inequality to verify it.

Suppose we want to determine the range of possible returns on an investment where the amount earned is no more than [latex]$200[/latex] above or below [latex]$600[/latex].

Solving algebraically:

  1. Write down the absolute value inequality: [latex]|x - 600|\le 200[/latex]
  2. Create two separate inequalities:
    • [latex]x - 600\le 200[/latex] (For the positive scenario)
    • [latex]x - 600\ge -200[/latex] (For the negative scenario)
  3. Solve for [latex]x[/latex] in both cases:
    • [latex]x \le 800[/latex]
    • [latex]x \ge 400[/latex] 
  4. Combine the solutions to state the final range: [latex]400\le x\le 800[/latex]

This means our returns would be between [latex]$400[/latex] and [latex]$800[/latex].

Solve [latex]|x - 1|\le 3[/latex].