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Understanding LCD and Solving Equations: A Step-by-Step Guide

This guide explains how to find the Least Common Denominator (LCD) of various equations and solve them step by step. It includes multiple examples such as finding LCD for expressions, solving rational equations, and checking for extraneous solutions. Through clear instructions, mathematical expressions are simplified, and important concepts are demonstrated, helping you strengthen your algebra skills. Ideal for students seeking to improve their understanding of rational expressions and equation solving techniques.

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Understanding LCD and Solving Equations: A Step-by-Step Guide

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  1. What is the LCD of the following equation? -1(x – 3) LCD = -2(x – 3) A 100

  2. After you multiply by the LCD, what is the resulting equation? LCD = 6x(x – 2) 2(x + 2)(x – 2) – 2x(6x) = x(x – 1)(x – 2) A 200

  3. Solve 9(x – 3) = 3(x + 5) 9x – 27 = 3x + 15 6x = 42 x = 7 A 300

  4. Solve + = + = LCD = 2x 1 + 2(2x – 3) = 2(2) 1 + 4x – 6 = 4 -5 + 4x = 4 4x = 9 x = 9/4 A 400

  5. Solve LCD = (a-1)(a+1) a(a – 1)(a + 1) + a2 – 5 = (a2 + a + 2)(a – 1) a(a2 – 1) + a2 – 5 = a3 – a2 + a2 – a + 2a – 2 a3 – a + a2 – 5 = a3 + a – 2 a2 – 2a – 3 = 0 (a – 3)(a + 1)= 0 a = 3 (-1 is extraneous) A 500

  6. What is the fully factored LCD of the related equation for this inequality? - < - < LCD = 8(x + 1)(x – 1) B 100

  7. When you solve the related equation below, the answers are x = 2 and x = 3. What is the final answer? x = 2, x = 3, x ≠ 0 Test x = -1 T x = 1 F x = 2.5 T x = 4 F x < 0 or 2 ≤ x ≤ 3 0 2 3 B 200

  8. Solve LCD = 6(x – 1) 6(x – 1) + 5(6) = 7(x – 1) 6x – 6 + 30 = 7x – 7 6x + 24 = 7x – 7 31 = x x ≠ 1 Test x = 0 T x = 2 F x = 32 T x < 1 or x ≥ 31 1 31 B 300

  9. Solve x2 – 7x + 12 = 0 (x – 3)(x – 4) = 0 x = 3, 4 x ≠ 5, 6 Test x = 0 T x = 3.5 F x = 4.5 T x = 5.5 F x = 7 T x < 3 or 4 ≤ x ≤ 5 or x > 6 3 6 5 4 B 400

  10. Solve LCD: 15(2x + 1)(x + 1) 15(x + 1) + 15(2x + 1) = 8(2x + 1)(x + 1) 15x + 15 + 30x + 15 = 16x2 + 24x + 8 0 = 16x2 + 21x – 22 0 = (16x + 11)(x – 2) x = -11/16 or 2 x ≠ -1/2 , -1 Test x = T x = 3.5 F x = 4.5 T x = 5.5 F x = 7 T x < 3 or 4 ≤ x ≤ 5 or x > 6 -1 2 B 500

  11. Which two x-values would you use to decompose the following rational expression? (2x – 1)(x + 3) Use x = ½ and x = -3 C 200

  12. What is the result after the first three steps of decomposing Factored LCD: (x – 1)(x + 2) = + 8x + 7 = A(x + 2) + B(x – 1) C 300

  13. Decompose x(x – 5) = + 15x – 35 = A(x – 5) + B(x) x = 0: 15(0) – 35 = A(0 – 5) + B(0) A = 7 x = 5: 15(5) – 35 = A(5 – 5) + B(5) B = -5 = + C 400

  14. Decompose (x – 2)(x + 2) = + 5x – 4 = A(x + 2) + B(x – 2) x = 2: 5(2) – 4 = A(2 + 2) + B(2 – 2) A = 1.5 x = -2: 5(-2) – 4 = A(-2 + 2) + B(-2 – 2) B = 3.5 = + C 500

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