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Introduction
Absolute Value helps you connect a mathematical rule to a result you can explain and check. Distance from zero is nonnegative. Evaluate inside absolute-value bars before taking the distance. This lesson belongs to algebra foundations and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in absolute value.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Locate signed numbers on a number line.
Previous lesson: Integers and Rational Numbers
Key mathematical vocabulary
- expression
- A mathematical phrase representing a value.
- solution
- A value making the original conditions true.
- input
- A value supplied to a rule.
- output
- The result of a rule at an allowed input.
- verify
- Check against the original conditions.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Absolute value measures distance
Distance from zero is nonnegative. Evaluate inside absolute-value bars before taking the distance.
Absolute value |x| is the distance of x from zero on a number line. Distances are nonnegative, so |x|=x when x≥0 and |x|=−x when x<0. Opposite numbers have the same absolute value.
Absolute value is not the instruction to erase every minus sign in an expression. Evaluate the entire quantity inside the bars first. For example, |2−7|=|−5|=5. An outside negative sign remains outside: −|−5|=−5.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. Distance counts units and cannot be negative.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. There are 56 units between −56 and 0.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
A position is −56 units from zero on a signed number line. What is its distance from zero?
- Distance counts units and cannot be negative.
- The absolute value of −56 is |−56|=56.
- The point +56 is the same distance from zero on the opposite side.
Answer: 56
Verify: There are 56 units between −56 and 0.
Avoid this mistake: Giving a negative distance: absolute value is nonnegative.
Example 2 · Basic
A position is −58 units from zero on a signed number line. What is its distance from zero?
- Distance counts units and cannot be negative.
- The absolute value of −58 is |−58|=58.
- The point +58 is the same distance from zero on the opposite side.
Answer: 58
Verify: There are 58 units between −58 and 0.
Avoid this mistake: Ignoring an outside negative sign.
Example 3 · Intermediate
A position is −60 units from zero on a signed number line. What is its distance from zero?
- Distance counts units and cannot be negative.
- The absolute value of −60 is |−60|=60.
- The point +60 is the same distance from zero on the opposite side.
Answer: 60
Verify: There are 60 units between −60 and 0.
Avoid this mistake: Changing signs before evaluating the entire inside expression.
Example 4 · Advanced
A position is −37 units from zero on a signed number line. What is its distance from zero?
- Distance counts units and cannot be negative.
- The absolute value of −37 is |−37|=37.
- The point +37 is the same distance from zero on the opposite side.
Answer: 37
Verify: There are 37 units between −37 and 0.
Avoid this mistake: Giving a negative distance: absolute value is nonnegative.
Example 5 · Challenge
A position is −39 units from zero on a signed number line. What is its distance from zero?
- Distance counts units and cannot be negative.
- The absolute value of −39 is |−39|=39.
- The point +39 is the same distance from zero on the opposite side.
Answer: 39
Verify: There are 39 units between −39 and 0.
Avoid this mistake: Ignoring an outside negative sign.
Common mistakes and corrections
- Giving a negative distance: absolute value is nonnegative.
- Ignoring an outside negative sign.
- Changing signs before evaluating the entire inside expression.
Real-world applications
- Compare distances above and below sea level. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Measure the size of a temperature change regardless of direction. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Distance from zero is nonnegative. Evaluate inside absolute-value bars before taking the distance.
Absolute value |x| is the distance of x from zero on a number line. Distances are nonnegative, so |x|=x when x≥0 and |x|=−x when x<0. Opposite numbers have the same absolute value.
Frequently asked questions
What should I know before absolute value?
Locate signed numbers on a number line.
How can I check my work?
There are 56 units between −56 and 0. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Giving a negative distance: absolute value is nonnegative. Ignoring an outside negative sign. Changing signs before evaluating the entire inside expression.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Compare distances above and below sea level.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Distance from zero is nonnegative. Evaluate inside absolute-value bars before taking the distance.
- Giving a negative distance: absolute value is nonnegative.
- Ignoring an outside negative sign.
- Changing signs before evaluating the entire inside expression.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
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Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials