Grade 11 · Mathematics
Quarter 3 · Week 6 · TEKS
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This lesson introduces Grade 11 students to formulating absolute value linear equations and solving both absolute value linear equations and inequalities. Students will learn how to represent real-world situations using absolute value expressions, translate these into equations or inequalities, and solve them accurately. The lesson aligns with TEKS standards for Algebra II and emphasizes mathematical reasoning, problem-solving, and application of absolute value concepts.
Students should understand the concept of absolute value as distance from zero on a number line, basic linear equations and inequalities, and properties of equality and inequality.
Absolute value represents the distance of a number from zero on the number line, always expressed as a non-negative value. For example, |x| means the distance of x from zero. This concept is crucial because absolute value equations and inequalities often model situations involving distances or magnitudes regardless of direction.
An absolute value linear equation involves an expression with absolute value set equal to another expression or number. For example, |2x - 3| = 7. To formulate such an equation, identify the quantity whose distance from zero is considered, then set the absolute value equal to a value or expression representing that distance. Real-world examples include distances from a point, tolerances in measurements, or deviations from a target value.
To solve |A| = B where B ≥ 0, rewrite as two separate linear equations: A = B and A = -B. Solve each equation separately. Check solutions in the original equation to avoid extraneous solutions. For example, solving |2x - 3| = 7 leads to 2x - 3 = 7 or 2x - 3 = -7, giving x = 5 or x = -2. Verify both solutions satisfy the original equation.
Inequalities involving absolute value, such as |A| < B or |A| > B, are solved by considering the definition of absolute value as distance. For |A| < B (where B > 0), rewrite as -B < A < B, a compound inequality. For |A| > B, rewrite as A < -B or A > B. Solve the resulting inequalities and express the solution set on a number line. For example, |x - 4| < 3 translates to 1 < x < 7.
- Students may forget to consider both positive and negative cases when solving absolute value equations. - They might treat absolute value inequalities like linear inequalities without rewriting them properly. - Some may overlook checking for extraneous solutions after solving. - Confusing the inequality direction when multiplying or dividing by negative numbers in compound inequalities.
Solve the absolute value equation |3x - 2| = 7.
Rewrite the equation as two separate linear equations: 3x - 2 = 7 and 3x - 2 = -7. Solve each: For 3x - 2 = 7, add 2 to both sides to get 3x = 9, then divide by 3 to get x = 3. For 3x - 2 = -7, add 2 to both sides to get 3x = -5, then divide by 3 to get x = -5/3. Both solutions satisfy the original equation.
x = 3 or x = -5/3
Solve the absolute value inequality |x + 4| < 5.
Rewrite the inequality as a compound inequality: -5 < x + 4 < 5. Subtract 4 from all parts: -9 < x < 1. The solution set includes all x values between -9 and 1, not including -9 and 1.
All x such that -9 < x < 1
Formulate an absolute value linear equation for the statement: The distance between a number x and 6 is 10.
The distance between x and 6 is represented by |x - 6|. Setting this equal to 10 gives the equation |x - 6| = 10.
|x - 6| = 10
Solve the absolute value equation |2x + 1| = 0.
Since absolute value equals zero only when the expression inside is zero, set 2x + 1 = 0. Solve for x: 2x = -1, x = -1/2.
x = -1/2
Solve the absolute value inequality |5 - 2x| ≥ 3.
Rewrite as two inequalities: 5 - 2x ≥ 3 or 5 - 2x ≤ -3. Solve each: For 5 - 2x ≥ 3, subtract 5: -2x ≥ -2, divide by -2 (reverse inequality): x ≤ 1. For 5 - 2x ≤ -3, subtract 5: -2x ≤ -8, divide by -2 (reverse inequality): x ≥ 4. Solution set is x ≤ 1 or x ≥ 4.
x ≤ 1 or x ≥ 4
Present a real-world scenario such as a delivery driver who must stay within 5 miles of a central location. Ask students how to represent the possible locations mathematically.
Discuss with a partner how to write an equation or inequality that represents the driver's allowed locations.
This scenario introduces absolute value as a measure of distance from a point, setting the stage for formulating absolute value equations or inequalities.
Provide students with expressions involving absolute value, such as |x - 3| = 4, and guide them to write two linear equations representing the positive and negative cases.
Work in pairs to formulate and solve the two linear equations derived from the absolute value equation.
Students explore how absolute value equations split into two cases, reinforcing the concept of distance being positive or negative relative to zero.
Explicitly teach the steps to solve absolute value linear equations and inequalities, including rewriting, solving, and checking solutions.
Take notes and ask clarifying questions. Solve example problems guided by the teacher.
Clear explanation of the algebraic process helps students understand the logic behind solving absolute value problems and avoid common errors.
Present real-world problems involving absolute value inequalities, such as tolerances in manufacturing or temperature ranges.
Formulate absolute value inequalities from the problems and solve them, then graph the solution sets on number lines.
Applying the concept to real situations deepens understanding and shows the usefulness of absolute value equations and inequalities.
Assign a set of problems including formulating, solving absolute value linear equations, and inequalities. Observe and provide feedback.
Independently solve the assigned problems and explain their reasoning in writing or orally.
Assessment of student understanding through problem-solving and explanation ensures mastery of the lesson objectives.
Work in pairs to formulate and solve the two linear equations derived from the absolute value equation.
What is the absolute value of -7?
7
Write an absolute value equation representing: The distance between a number x and 3 is 8.
|x - 3| = 8
Solve the absolute value equation |x - 5| = 9.
x = 14 or x = -4
Solve the absolute value inequality |2x + 3| < 7.
-5 < x < 2
Solve the absolute value inequality |4 - x| ≥ 6.
x ≤ -2 or x ≥ 10
Use number lines and manipulatives to visualize absolute value as distance. Provide step-by-step guided examples with scaffolding. Allow use of calculators for arithmetic to focus on conceptual understanding.
Use visual aids like number lines and diagrams to explain absolute value. Pre-teach vocabulary such as 'distance', 'absolute value', 'inequality'. Encourage peer discussion and use of sentence frames for explaining reasoning.
Challenge students with absolute value equations involving variables on both sides. Explore solving absolute value inequalities with quadratic expressions inside. Investigate real-world modeling problems requiring formulation and solution of absolute value equations.
Ask students to find examples of absolute value in everyday life, such as temperature differences, distances, or tolerances, and write equations or inequalities to represent those situations.
In this lesson, students learned that absolute value represents distance from zero and how to formulate absolute value linear equations from real-world contexts. They practiced solving absolute value linear equations by considering two cases and solving absolute value inequalities by rewriting them as compound inequalities. Students also learned to check solutions and interpret solution sets on number lines. These skills enable them to model and solve problems involving distances and tolerances effectively.
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