Intermediate Algebra Problems With Answers - Sample 11
Simplify Algebraic Expressions by Removing Brackets

This page features a collection of intermediate algebra problems focused on simplifying algebraic expressions. Students will practice using the distributive property to expand expressions and eliminate brackets. Fully worked step-by-step solutions are provided for each problem below.

Practice Problems & Step-by-Step Solutions

Section 1: Problems with One Pair of Brackets

Problems 1(a) to 1(h): Single Pair of Brackets

Problems: Simplify each expression by removing the brackets and grouping like terms:

  1. \( 2(3x + 4) + 6 \)
  2. \( -4(x + 3) - 7 \)
  3. \( -2(x - 2) + 4 \)
  4. \( -2(-x - 8) + 4 \)
  5. \( -\dfrac{3}{2}(-x - 8) + 4 \)
  6. \( -1.2(-2.2x - 1.7) + 0.2 \)
  7. \( 5(2x + 3y - 4) - x + 2y + 6 \)
  8. \( x(2x + 3y - 4) - x^2 + 4xy - 12 \)

Solutions:

  1. \( 2(3x + 4) + 6 \)
    \(= 6x + 8 + 6 \)
    \(= 6x + 14 \)
  2. \( -4(x + 3) - 7 \)
    \(= -4x - 12 - 7 \)
    \(= -4x - 19 \)
  3. \( -2(x - 2) + 4 \)
    \(= -2x + 4 + 4 \)
    \(= -2x + 8 \)
  4. \( -2(-x - 8) + 4 \)
    \(= 2x + 16 + 4 \)
    \(= 2x + 20 \)
  5. \( -\dfrac{3}{2}(-x - 8) + 4 \)
    \(= \dfrac{3}{2}x + 12 + 4 \)
    \(= \dfrac{3}{2}x + 16 \)
  6. \( -1.2(-2.2x - 1.7) + 0.2 \)
    \(= 2.64x + 2.04 + 0.2 \)
    \(= 2.64x + 2.24 \)
  7. \( 5(2x + 3y - 4) - x + 2y + 6 \)
    \(= 10x + 15y - 20 - x + 2y + 6 \)
    \(= 9x + 17y - 14 \)
  8. \( x(2x + 3y - 4) - x^2 + 4xy - 12 \)
    \(= 2x^2 + 3xy - 4x - x^2 + 4xy - 12 \)
    \(= x^2 + 7xy - 4x - 12 \)

Section 2: Problems with Two Pairs of Brackets

Problems 2(a) to 2(e): Two Pairs of Brackets

Problems: Simplify each expression by removing the brackets and grouping like terms:

  1. \( 3(x - 5) + 6(x + 3) \)
  2. \( -5(2x + 3) - 2(x - 3) \)
  3. \( (x - 2)(x + 3) + 8 \)
  4. \( (x - 2)^2 + 2(x - 4) \)
  5. \( y(x + 3) - x(2y + 4) - 7x - 8y + 2 \)

Solutions:

  1. \( 3(x - 5) + 6(x + 3) \)
    \(= 3x - 15 + 6x + 18 \)
    \(= 9x + 3 \)
  2. \( -5(2x + 3) - 2(x - 3) \)
    \(= -10x - 15 - 2x + 6 \)
    \(= -12x - 9 \)
  3. \( (x - 2)(x + 3) + 8 \)
    \(= x^2 + 3x - 2x - 6 + 8 \)
    \(= x^2 + x + 2 \)
  4. \( (x - 2)^2 + 2(x - 4) \)
    \(= x^2 - 4x + 4 + 2x - 8 \)
    \(= x^2 - 2x - 4 \)
  5. \( y(x + 3) - x(2y + 4) - 7x - 8y + 2 \)
    \(= yx + 3y - 2xy - 4x - 7x - 8y + 2 \)
    \(= -xy - 11x - 5y + 2 \)

Section 3: Problems with Nested Brackets

Problems 3(a) to 3(d): Nested Brackets

Problems: Simplify each expression by removing the brackets and grouping like terms:

  1. \( -2 \left( 3(x + 2) + 4 \right) - 8 \)
  2. \( 4 \left( -3x - 2(x - 9) \right) - 7(x - 2) \)
  3. \( 2 - \left( 3 - \left( -5(3x + 1) + 4 \right) \right) \)
  4. \( \left( (3 - x)(x + 2) + (-x + 4)(7x + 2) - (x - y)(2x - y) \right) - 3x^2 - 7x + 5 \)

Solutions:

  1. \( -2 \left( 3(x + 2) + 4 \right) - 8 \)
    \(= -2(3x + 6 + 4) - 8 \)
    \(= -2(3x + 10) - 8 \)
    \(= -6x - 20 - 8 \)
    \(= -6x - 28 \)
  2. \( 4 \left( -3x - 2(x - 9) \right) - 7(x - 2) \)
    \(= 4(-3x - 2x + 18) - 7x + 14 \)
    \(= 4(-5x + 18) - 7x + 14 \)
    \(= -20x + 72 - 7x + 14 \)
    \(= -27x + 86 \)
  3. \( 2 - \left( 3 - \left( -5(3x + 1) + 4 \right) \right) \)
    \(= 2 - \left( 3 - (-15x - 5 + 4) \right) \)
    \(= 2 - \left( 3 - (-15x - 1) \right) \)
    \(= 2 - (3 + 15x + 1) \)
    \(= 2 - (15x + 4) \)
    \(= 2 - 15x - 4 \)
    \(= -15x - 2 \)
  4. \( \left[ (3 - x)(x + 2) + (-x + 4)(7x + 2) - (x - y)(2x - y) \right] - 3x^2 - 7x + 5 \)
    \(= \left[ 3x + 6 - x^2 - 2x + (-7x^2 - 2x + 28x + 8) - (2x^2 - xy - 2yx + y^2) \right] - 3x^2 - 7x + 5 \)
    \(= \left[ -x^2 + x + 6 - 7x^2 + 26x + 8 - 2x^2 + xy + 2yx - y^2 \right] - 3x^2 - 7x + 5 \)
    \(= \left[ (-x^2 - 7x^2 - 2x^2) + (xy + 2yx) + (x + 26x) + 6 + 8 - y^2 \right] - 3x^2 - 7x + 5 \)
    \(= -10x^2 + 3xy + 27x - y^2 + 14 - 3x^2 - 7x + 5 \)
    \(= -13x^2 + 3xy - y^2 + 20x + 19 \)

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