ACT Math Pre-Algebra Absolute Value Solving Equations and Inequalities

Pre-Algebra — Absolute Value: Solving Equations and Inequalities

Difficulty Level: Intermediate

What This Is and Why It Matters for the ACT

Absolute value equations and inequalities appear in the Math section of the ACT, often in the no-calculator section. They test your ability to solve and graph equations with absolute values. Be prepared to see at least one question on this topic in every ACT Math test.

Key Concepts (What You Must Know)

• Absolute value is the distance of a number from zero on the number line.
• The formula for absolute value is |x| = x if x ≥ 0 and |x| = -x if x < 0.
• To solve an absolute value equation, you must consider both the positive and negative cases.
• Be aware of the following vocabulary:
  • Inequality: a statement that two expressions are not equal.
  • Solution: a value that makes an equation or inequality true.

Step-by-Step Strategy for This Topic

1. Read the question carefully and identify the type of absolute value equation or inequality.
2. If it's an equation, consider both the positive and negative cases.
3. If it's an inequality, determine the direction of the inequality sign and solve for the variable.
4. Check your work by plugging in your solution to the original equation or inequality.
5. Time management tip: spend no more than 1 minute and 30 seconds on each question.

How It’s Tested on the ACT

Math questions on absolute value often look like this: * A multiple-choice question with five answer choices, such as:
+ What is the solution to the equation |x| = 5?
+ A, 5
+ B, -5
+ C, 0
+ D, 2
+ E, -2 * Common distractors include:
+ Forgetting to consider both the positive and negative cases.
+ Misreading the inequality sign.
+ Not checking your work.

Common Mistakes & Exam Traps

• The mistake: Forgetting to consider both the positive and negative cases.
  • Why it happens: Rushing through the question or misreading the equation.
  • How to avoid it: Take a deep breath and read the question carefully before starting to solve.
• The mistake: Misreading the inequality sign.
  • Why it happens: Not paying attention to the direction of the inequality sign.
  • How to avoid it: Double-check the inequality sign before solving.
• The mistake: Not checking your work.
  • Why it happens: Rushing through the question or being confident in your answer.
  • How to avoid it: Take a minute to plug in your solution to the original equation or inequality.

Practice Questions (3-5 questions)

Question 1: What is the solution to the equation |x| = 3? Options: A, 3 B, -3
C, 0
D, 2
E, -2
Answer: B, -3
Explanation: To solve the equation, you must consider both the positive and negative cases. Since |x| = 3, x must be either 3 or -3. Therefore, the solution is -3.

Question 2: What is the solution to the inequality |x| > 2? Options: A, x > 2 B, x < -2
C, x > -2
D, x < 2
E, x = 2
Answer: B, x < -2
Explanation: To solve the inequality, you must determine the direction of the inequality sign. Since |x| > 2, x must be either less than -2 or greater than 2. Therefore, the solution is x < -2.

Quick Reference Card (60-Second Summary)

• |x| = x if x ≥ 0
• |x| = -x if x < 0
• Consider both the positive and negative cases when solving absolute value equations.
• Check your work by plugging in your solution to the original equation or inequality.
• Double-check the inequality sign before solving.

If You Get Stuck on Test Day

• If you don't know the answer, eliminate any obviously incorrect options and make an educated guess.
• Spend no more than 1 minute and 30 seconds on each question.
• If you're stuck, move on to the next question and come back to it later.

Related ACT Topics

• Linear Equations and Inequalities: This topic is closely related to absolute value equations and inequalities. Be prepared to see questions that combine both topics.
• Quadratic Equations: Quadratic equations often involve absolute value expressions. Be prepared to see questions that require you to solve quadratic equations with absolute value.
• Graphing: Graphing is an important skill for solving absolute value equations and inequalities. Be prepared to see questions that require you to graph absolute value functions.