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Chapter 1

Chapter 1. Section 5 Absolute Value Equations and Inequalities. >. –. Absolute Value Equations and Inequalities. ALGEBRA 2 LESSON 1-5. (For help, go to Lessons 1-3 and 1-4.). Solve each equation. 1. 5( x – 6) = 40. 2. 5 b = 2(3 b – 8). 3. 2 y + 6 y = 15 – 2 y + 8.

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Chapter 1

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  1. Chapter 1 Section 5 Absolute Value Equations and Inequalities

  2. > – Absolute Value Equations and Inequalities ALGEBRA 2 LESSON 1-5 (For help, go to Lessons 1-3 and 1-4.) Solve each equation. 1. 5(x – 6) = 40 2. 5b = 2(3b – 8) 3. 2y + 6y = 15 – 2y + 8 Solve each inequality. 4. 4x + 8 > 20 5. 3a – 2 a + 6 6. 4(t – 1) < 3t + 5

  3. > > > > – – – – Absolute Value Equations and Inequalities ALGEBRA 2 LESSON 1-5 1. 5(x – 6) = 40 2. 5b = 2(3b – 8) = 5b = 6b – 16 x – 6 = 8 –b = –16 x = 14 b = 16 Solutions 40 5 5(x – 6) 5 3. 2y + 6y = 15 – 2y + 8 4. 4x + 8 > 20 2y + 6y + 2y = 15 + 8 4x > 12 10y = 23 x > 3 y = 2.3 5. 3a – 2 a + 6 6. 4(t – 1) < 3t + 5 3a – a 6 + 2 4t – 4 < 3t + 5 2a 8 4t – 3t < 5 + 4 a 4 t < 9

  4. Absolute Value • Absolute Value – the distance from zero a number is on the number line – it is always positive • Symbol : │x│ • Definition: • If x is positive ( x > 0) then │x│ = x • If x is negative ( x < 0) then │x │ = -x • Absolute Value Equations have a possibility of two solutions • This is because the value inside the │ │ can equal either the negative or the positive of the value on the other side of the equal sign • Always isolate the absolute value expression on one side of the equal sign before breaking the problem into two pieces.

  5. –3x = –9 –3x = –21 Subtract 15 from each side of both equations. Check: |15 – 3x| = 6 |15 – 3x| = 6 |15 – 3(3)| 6 |15 – 3(7)| 6 |6| 6 |–6| 6 6 = 6 6 = 6 Absolute Value Equations and Inequalities Solve |15 – 3x| = 6. |15 – 3x| = 6 15 – 3x = 6 or 15 – 3x = –6 The value of 15 – 3x can be 6 or –6 since |6| and |–6| both equal 6. x = 3 or x = 7 Divide each side of both equations by –3.

  6. Try This Problem │3x + 2 │ = 7 3x + 2 = 7 or 3x + 2 = -7 3x = 5 or 3x = -9 x = 5/3 or x = -3 Check your answer by plugging it back into the equation. .

  7. 9 2 |x + 9| = Divide each side by –2. 9 2 9 2 x + 9 = or x + 9 =– Rewrite as two equations. Check: 4 – 2 |x + 9| = –5 4 – 2|x + 9| = –5 4 – 2 |–4.5 + 9| –5 4 – 2 |–13.5 + 9| –5 4 – 2 |4.5| –5 4 – 2 |–4.5| –5 4 – 2 (4.5) –5 4 – 2 (4.5) –5 –5 = –5 –5 = –5 Absolute Value Equations and Inequalities Solve 4 – 2|x + 9| = –5. 4 – 2|x + 9| = –5 –2|x + 9| = –9 Add –4 to each side. x = –4.5 or x =–13.5 Subtract 9 from each side of both equations.

  8. Try This Problem Solve 2│3x - 1 │ + 5 = 33. 2 │3x - 1 │ = 28 │3x – 1│ = 14 3x – 1 = 14 or 3x – 1 = -14 3x = 15 or 3x = -14 x = 5 or x = -8/3 Check the solutions by plugging them into the original problem. They both work. 

  9. Extraneous Solutions • Extraneous solution – a solution of an equation derived from an original equation that is not a solution of the original equation • This is why we MUST check all answers to see if they work in the original problem.

  10. 2x + 5 = 3x + 4 5 = x + 4 1 = x Check: 2(1) + 5 = 3(1) + 4 2 + 5 = 3 + 4 7 = 7 2x + 5 = -(3x + 4) 2x + 5 = -3x – 4 5x + 5 = -4 5x = -9 x = -9/5 Check: │2(-9/5) + 5 │ = 3(-9/5) + 4 │-18/5 + 25/5 │ = -27/5 + 20/5 │7/5 │ = -7/5 │7/5 │ ≠ -7/5 Solve │2x + 5 │ = 3x + 4 The only solution is x = 1.

  11. │2x + 3 │ = 3x + 2 2x + 3 = 3x + 2 or 2x + 3 = -3x – 2 3 = x + 2 or 5x + 3 = -2 1 = x or 5x = -5 1 = x or x = -1 Check: The solution is x = 1 │x │ = x – 1 x = x – 1 or x = -x + 1 0 = -1 or 2x = 1 x = ½ Check: There is No Solution. Try These ProblemsSolve and check for extraneous solutions.

  12. Homework • Practice 1.5 #13-24 all

  13. Absolute Value Inequalities Let k represent a positive real number • │x │ ≥ k is equivalent to x ≤ -k or x ≥ k • │x │ ≤ k is equivalent to -k ≤ x ≤ k • Remember to isolate the absolute value before rewriting the problem with two inequalities

  14. 2x < 2 2x > 8 Solve for x. Solve |2x – 5| > 3. Graph the solution. |2x – 5| > 3 2x – 5 < –3 or 2x – 5 > 3 Rewrite as a compound inequality. x < 1 or x > 4

  15. Try This Problem Solve │2x - 3 │ > 7 2x – 3 > 7 or 2x – 3 < -7 2x > 10 or 2x < -4 x > 5 or x < -2

  16. < < < < < – – – – – > > > – – – –2|x + 1| + 5 –3 –2|x + 1| –8 Isolate the absolute value expression. Subtract 5 from each side. |x + 1| 4 Divide each side by –2 and reverse the inequality. –4 x + 1 4 Rewrite as a compound inequality. –5 x 3 Solve for x. Solve –2|x + 1| + 5 –3. Graph the solution.

  17. Try This Problem Solve |5z + 3| - 7 < 34. Graph the solution. |5z + 3| -7 < 34 |5z + 3| < 41 -41 < 5z + 3 < 41 -44 < 5z < 38 -44/5 < z < 38/5 -8 4/5 < z < 7 3/5

  18. Ranges in Measurement • Absolute value inequalities and compound inequalities can be used to specify an allowable range in measurement. • Tolerance – the difference between a measurement and its maximum and minimum allowable values • Equals one half (½) the difference between the max and min values

  19. Tolerance For example, if a manufacturing specification calls for a dimension d of 10 cm with a tolerance of 0.1 cm, then the allowable difference between d and 10 is less than or equal to 0.1 • |d - 10 | ≤ 0.1 absolute value inequality • d – 10 ≤ 0.1 and d – 10 ≥-0.1 equivalent compound inequality • -0.1 ≤ d – 10 ≤ 0.1 equivalent compound inequality • 9.9 ≤ d ≤ 10.1 simplified compound inequality

  20. < < < – – – 8.9 – 8.5 2 0.4 2 = = 0.2 Find the tolerance. 8.9 + 8.5 2 17.4 2 = = 8.7 Find the average of the maximum andminimum values. –0.2 A – 8.7 0.2 Write an inequality. |A – 8.7| 0.2 Rewrite as an absolute value inequality. The area A in square inches of a square photo is required to satisfy 8.5 ≤A≤ 8.9. Write this requirement as an absolute value inequality.

  21. Try This Problem The specification for the circumference C in inches of a basketball for junior high school is 27.75 ≤ C ≤ 30. Write the specification as an absolute value inequality. Find the tolerance. Find the average from min and max values. Write the absolute value inequality.

  22. Homework • Practice 1.5 # 1 – 12, 25 – 30 • Watch the units on the back page • Make sure all units are inches, centimeters, meters, etc…

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