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Equations with Absolute Value

Equations with Absolute Value. Equations with Absolute Value Essential Question. M8A1c. Solve algebraic equations or inequalities in one variable, including those involving absolute values. How does absolute value affect the solution of an equation?. 4 units 4 units.

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Equations with Absolute Value

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  1. Equations with Absolute Value

  2. Equations with Absolute Value Essential Question M8A1c. Solve algebraic equations or inequalities in one variable, including those involving absolute values. How does absolute value affect the solution of an equation?

  3. 4 units 4 units –5 –4 –3 –2 –1 0 1 2 3 4 5 A number’s absolute value is its distance from 0 on a number line. Absolute value is always positive because distance is always positive. “The absolute value of –4” is written as |–4|. Opposites have the same absolute value. |–4| = 4 |4| = 4

  4. 3 units –5 –4 –3 –2 –1 0 1 2 3 4 5 Additional Example 3: Simplifying Absolute-Value Expressions Simplify each expression. A. |–3| –3 is 3 units from 0, so |–3| = 3. B. |17 – 6| |17 – 6| = |11| Subtract first: 17 – 6 = 11. Then find the absolute value: 11 is 11 units from 0. = 11

  5. Additional Example 3: Simplifying Absolute-Value Expressions Simplify each expression. C. |–8| + |–5| Find the absolute values first: –8 is 8 units from 0. –5 is 5 units from 0. Then add. |–8| + |–5| = 8 + 5 = 13 D. |5 + 1| + |8 – 6| |5 + 1| + |8 – 6| = |6| + |2| 5 + 1 = 6, 8 – 6 = 2. 6 is 6 units from 0, 2 is 2 units from 0. Add. = 6 + 2 = 8

  6. 5 units –5 –4 –3 –2 –1 0 1 2 3 4 5 Check It Out! Example 3 Simplify each expression. A. |–5| –5 is 5 units from 0, so |–5| = 5. B. |12 – 4| |12 – 4| = |8| Subtract first: 12 – 4 = 8. Then find the absolute value: 8 is 8 units from 0. = 8

  7. Check It Out! Example 3 Simplify each expression. C. |–2| + |–9| Find the absolute values first: –2 is 2 units from 0. –9 is 9 units from 0. Then add. |–2| + |–9| = 2 + 9 = 11 D. |3 + 1| + |9 – 2| |3 + 1| + |9 – 2| = |4| + |7| 3 + 1 = 4, 9 – 2 = 7. 4 is 4 units from 0, 7 is 7 units from 0. Add. = 4 + 7 = 11

  8. Symbol │x│ The distance x is from 0 on the number line. Always positive Ex: │-3│=3 Absolute Value (of x) -4 -3 -2 -1 0 1 2

  9. You can solve some absolute-value equations using mental math. For instance, you learned that the equation |x| 8 has two solutions: 8 and 8. To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative.

  10. First, isolate the absolute-value expression on the left side of the equality. │ ax + b │= c Rewrite as Positive Case ax + b = c and solve for x. Rewrite as Negative case ax + b = -c and solve for x. Solve 6x – 3 = 15 Solve x – 2 = 5 Graphic Organizer by Dale Graham and Linda Meyer Thomas County Central High School; Thomasville GA

  11. Solving an Absolute-Value Equation The expressionx  2 can be equal to 5 or 5. x  2IS NEGATIVE | x  2 |  5 x  2IS POSITIVE x  2 IS POSITIVE x  2 IS POSITIVE x  2 IS NEGATIVE x  2  5 | x  2 |  5 | x  2 |  5 x  3 x  2  5 x  2  5 x  2  5 x  2  5 CHECK x  7 x  7 Solve | x  2 |  5 Solve | x  2 |  5 SOLUTION The expressionx  2can be equal to5or5. x  2 IS POSITIVE x  2 IS NEGATIVE | x  2 |  5 | x  2 |  5 x  2  5 x  2  5 x  7 x  3 The equation has two solutions: 7 and –3. | 7  2 |  | 5 |  5 |3  2 |  | 5 |  5

  12. 6x-3 = 15 or 6x-3 = -15 6x = 18 or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions! Ex: Solve 6x-3 = 15

  13. Solving an Absolute-Value Equation Isolate theabsolute value expressionon one side of the equation. 2x  7 IS NEGATIVE 2x  7 IS POSITIVE 2x  7 IS NEGATIVE 2x  7 IS POSITIVE 2x  7 IS POSITIVE | 2x  7 |  5  4 | 2x  7 |  5  4 | 2x  7 |  5  4 | 2x  7 |  9 | 2x  7 |  9 | 2x  7 |  9 2x  7  9 2x  7  +9 2x  7  9 2x  7  +9 2x  7  +9 2x  2 2x  16 2x  16 x  8 x  1 x  1 TWO SOLUTIONS x  8 x  8 Solve | 2x  7 |  5  4 Solve | 2x  7 |  5  4 SOLUTION Isolate theabsolute value expressionon one side of the equation. 2x  7 IS POSITIVE 2x  7 IS NEGATIVE | 2x  7 |  5  4 | 2x  7 |  5  4 | 2x  7 |  9 | 2x  7 |  9 2x  7  +9 2x  7  9 2x  16 2x  2 x  8 x  1

  14. Get the absolute value part by itself first! 2x+7 = 11 Now split into 2 parts. 2x+7 = 11 or 2x+7 = -11 2x = 4 or 2x = -18 x = 2 or x = -9 Check the solutions. Ex: Solve 2x + 7 -3 = 8

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