{"id":9877,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9877"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"reducing-equation","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/reducing-equation\/","title":{"rendered":"Reducing Equation"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Algebra – 1<\/strong><\/p>\n

Chapter: <\/strong>Reducing Equations<\/strong><\/p>\n

Reference: – Introduction to Reducing Equations, Equations with Parentheses, Equations with Fractions, Equations with Decimals, Cross Multiplication Method, Equations with Variables in Denominator, Equations Containing Nested Parentheses, Simplifying Before Solving, Converting Complex to Standard Form, Solved Examples, Odd-One-Out Problems, Common Mistakes, Practice Grid<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • Reducing Complex Equations to Simple form<\/em><\/li>\n
  • Removing Parentheses, Fractions, and Decimals<\/em><\/li>\n
  • Cross Multiplication for Fraction Equation<\/em><\/li>\n
  • Simplifying before Solving<\/em><\/li>\n<\/ul>\n

    \u00a0<\/p>\n

    Introduction to Reducing Equations<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    Reducing an equation means simplifying it from a more complex form to the standard linear form\u00a0ax + b = c\u00a0or\u00a0ax + b = cx + d. Many equations do not appear linear at first glance but can be “reduced” to linear equations through simplification.<\/p>\n

    When we reduce an equation, we essentially ask:<\/p>\n

    “How can I simplify this equation to the basic form I already know how to solve?”<\/p>\n

    Once reduced, the equation can be solved using standard techniques (variable on one side or both sides).<\/p>\n

    Importance of Reducing Equations<\/u><\/strong><\/p>\n

      \n
    • Enables solving of real-world problems that aren’t initially in simple form<\/li>\n
    • Builds algebraic manipulation skills<\/li>\n
    • Essential for advanced topics like rational equations and literal equations<\/li>\n
    • Teaches systematic simplification<\/li>\n<\/ul>\n

      Example<\/strong><\/p>\n

      Equation:<\/strong>\u00a02(x + 3) + 5 = 3(x – 1)
      \nStep 1 (Reduce):<\/strong>\u00a02x + 6 + 5 = 3x – 3 \u2192 2x + 11 = 3x – 3
      \nStep 2 (Solve):<\/strong>\u00a02x + 11 = 3x – 3 \u2192 11 + 3 = 3x – 2x \u2192 x = 14<\/p>\n

      So, the original complex equation\u00a0reduces\u00a0to x = 14.<\/p>\n

      \u00a0<\/p>\n

      Subtopics<\/u><\/strong><\/p>\n

      1. Why Reduce Equations?<\/strong><\/p>\n

      Complex equations are difficult to solve directly. Reducing converts them into familiar forms, making the solution process systematic and error-free.<\/p>\n

      2. Common Reduction Techniques<\/strong><\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Technique<\/p>\n<\/td>\n

      		\n

      Used For<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Remove parentheses<\/p>\n<\/td>\n

      		\n

      Expressions like a(bx + c)<\/p>\n<\/td>\n

      		\n

      Distribute<\/p>\n<\/td>\n<\/tr>\n

      \n

      Clear fractions<\/p>\n<\/td>\n

      		\n

      Equations with denominators<\/p>\n<\/td>\n

      		\n

      Multiply by LCM<\/p>\n<\/td>\n<\/tr>\n

      \n

      Clear decimals<\/p>\n<\/td>\n

      		\n

      Equations with decimal numbers<\/p>\n<\/td>\n

      		\n

      Multiply by power of 10<\/p>\n<\/td>\n<\/tr>\n

      \n

      Cross multiply<\/p>\n<\/td>\n

      		\n

      Equations of form a\/b = c\/d<\/p>\n<\/td>\n

      		\n

      Multiply both sides by denominators<\/p>\n<\/td>\n<\/tr>\n

      \n

      Simplify like terms<\/p>\n<\/td>\n

      		\n

      Equations with multiple terms<\/p>\n<\/td>\n

      		\n

      Combine constants and variable terms<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      \u00a0<\/p>\n

      Reducing Equations with Parentheses<\/strong><\/p>\n

      Definition<\/u><\/strong><\/p>\n

      Equations containing parentheses are reduced by\u00a0distributing\u00a0the term outside the parentheses to each term inside.<\/p>\n

      General Rule:<\/strong>\u00a0a(bx + c) = abx + ac<\/p>\n

      Example 1:<\/strong>\u00a0Solve 4(x + 3) = 2(x + 7)<\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      4(x + 3) = 2(x + 7)<\/p>\n<\/td>\n<\/tr>\n

      \n

      Distribute<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      4x + 12 = 2x + 14<\/p>\n<\/td>\n<\/tr>\n

      \n

      Subtract 2x<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      2x + 12 = 14<\/p>\n<\/td>\n<\/tr>\n

      \n

      Subtract 12<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      2x = 2<\/p>\n<\/td>\n<\/tr>\n

      \n

      Divide by 2<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      x = 1<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Example 2:<\/strong>\u00a0Solve 3(2x – 5) + 4 = 2(3x – 4)<\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      3(2x – 5) + 4 = 2(3x – 4)<\/p>\n<\/td>\n<\/tr>\n

      \n

      Distribute<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      6x – 15 + 4 = 6x – 8<\/p>\n<\/td>\n<\/tr>\n

      \n

      Simplify LHS<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      6x – 11 = 6x – 8<\/p>\n<\/td>\n<\/tr>\n

      \n

      Subtract 6x<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      -11 = -8<\/p>\n<\/td>\n<\/tr>\n

      \n

      Result<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      False \u2192 No solution<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Example 3:<\/strong>\u00a0Solve 5x – 2(3x + 1) = 4<\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      5x – 2(3x + 1) = 4<\/p>\n<\/td>\n<\/tr>\n

      \n

      Distribute (careful with minus)<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      5x – 6x – 2 = 4<\/p>\n<\/td>\n<\/tr>\n

      \n

      Simplify LHS<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      -x – 2 = 4<\/p>\n<\/td>\n<\/tr>\n

      \n

      Add 2<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      -x = 6<\/p>\n<\/td>\n<\/tr>\n

      \n

      Multiply by -1<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      x = -6<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      \u00a0<\/p>\n

      Reducing Equations with Fractions<\/strong><\/p>\n

      Definition<\/u><\/strong><\/p>\n

      Equations containing fractions are reduced by multiplying both sides by the\u00a0LCM (Least Common Multiple)\u00a0of all denominators. This clears the fractions.<\/p>\n

      Example 1:<\/strong>\u00a0Solve x\/2 + x\/3 = 10<\/p>\n

      \n\n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      x\/2 + x\/3 = 10<\/p>\n<\/td>\n<\/tr>\n

      \n

      LCM of 2 and 3 = 6<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\u00a0<\/td>\n<\/tr>\n
      \n

      Multiply both sides by 6<\/p>\n<\/td>\n

      		\n

      6(x\/2) + 6(x\/3) = 60<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Simplify<\/p>\n<\/td>\n

      		\n

      3x + 2x = 60<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Combine<\/p>\n<\/td>\n

      		\n

      5x = 60<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Divide<\/p>\n<\/td>\n

      		\n

      x = 12<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Example 2:<\/strong>\u00a0Solve (x + 1)\/3 + (x – 1)\/2 = 1<\/p>\n

      \n\n\n\n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      (x+1)\/3 + (x-1)\/2 = 1<\/p>\n<\/td>\n<\/tr>\n

      \n

      LCM of 3 and 2 = 6<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\u00a0<\/td>\n<\/tr>\n
      \n

      Multiply by 6<\/p>\n<\/td>\n

      		\n

      6\u00d7(x+1)\/3 + 6\u00d7(x-1)\/2 = 6\u00d71<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Simplify<\/p>\n<\/td>\n

      		\n

      2(x+1) + 3(x-1) = 6<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Distribute<\/p>\n<\/td>\n

      		\n

      2x + 2 + 3x – 3 = 6<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Combine<\/p>\n<\/td>\n

      		\n

      5x – 1 = 6<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Add 1<\/p>\n<\/td>\n

      		\n

      5x = 7<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Divide<\/p>\n<\/td>\n

      		\n

      x = 7\/5 = 1.4<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Example 3:<\/strong>\u00a0Solve (2x – 1)\/3 = (x + 2)\/4<\/p>\n

      \n\n\n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      (2x – 1)\/3 = (x + 2)\/4<\/p>\n<\/td>\n<\/tr>\n

      \n

      LCM of 3 and 4 = 12<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\u00a0<\/td>\n<\/tr>\n
      \n

      Multiply by 12<\/p>\n<\/td>\n

      		\n

      4(2x – 1) = 3(x + 2)<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Distribute<\/p>\n<\/td>\n

      		\n

      8x – 4 = 3x + 6<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Subtract 3x<\/p>\n<\/td>\n

      		\n

      5x – 4 = 6<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Add 4<\/p>\n<\/td>\n

      		\n

      5x = 10<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Divide<\/p>\n<\/td>\n

      		\n

      x = 2<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      \u00a0<\/p>\n

      Cross Multiplication Method<\/strong><\/p>\n

      Definition<\/u><\/strong><\/p>\n

      When an equation is of the form\u00a0a\/b = c\/d\u00a0(two fractions equal to each other), cross multiplication is the most efficient method.<\/p>\n

      Formula:<\/strong>\u00a0If a\/b = c\/d, then a \u00d7 d = b \u00d7 c<\/p>\n

      Example 1:<\/strong>\u00a0Solve 3\/(x + 2) = 4\/(x – 1)<\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      3\/(x+2) = 4\/(x-1)<\/p>\n<\/td>\n<\/tr>\n

      \n

      Cross multiply (assume denominators \u2260 0)<\/p>\n<\/td>\n

      		\n

      3(x – 1) = 4(x + 2)<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Distribute<\/p>\n<\/td>\n

      		\n

      3x – 3 = 4x + 8<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Subtract 3x<\/p>\n<\/td>\n

      		\n

      -3 = x + 8<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Subtract 8<\/p>\n<\/td>\n

      		\n

      -11 = x<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Check denominators:<\/strong>\u00a0x+2 = -11+2 = -9 \u2260 0; x-1 = -12 \u2260 0 \u2713<\/p>\n

      Example 2:<\/strong>\u00a0Solve (x + 2)\/5 = 3\/4<\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      (x+2)\/5 = 3\/4<\/p>\n<\/td>\n<\/tr>\n

      \n

      Cross multiply<\/p>\n<\/td>\n

      		\n

      4(x + 2) = 15<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Distribute<\/p>\n<\/td>\n

      		\n

      4x + 8 = 15<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Subtract 8<\/p>\n<\/td>\n

      		\n

      4x = 7<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Divide<\/p>\n<\/td>\n

      		\n

      x = 7\/4 = 1.75<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      \u00a0<\/p>\n

      Reducing Equations with Decimals<\/strong><\/p>\n

      Definition<\/u><\/strong><\/p>\n

      Multiply both sides by a power of 10 (10, 100, 1000, etc.) to convert decimals to integers.<\/p>\n

      Example 1:<\/strong>\u00a0Solve 0.5x + 0.2 = 1.3<\/p>\n

      \n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      0.5x + 0.2 = 1.3<\/p>\n<\/td>\n<\/tr>\n

      \n

      Multiply by 10<\/p>\n<\/td>\n

      		\n

      5x + 2 = 13<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Subtract 2<\/p>\n<\/td>\n

      		\n

      5x = 11<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Divide<\/p>\n<\/td>\n

      		\n

      x = 2.2<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Example 2:<\/strong>\u00a0Solve 1.2x – 0.7 = 2x – 1.5<\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      1.2x – 0.7 = 2x – 1.5<\/p>\n<\/td>\n<\/tr>\n

      \n

      Multiply by 10<\/p>\n<\/td>\n

      		\n

      12x – 7 = 20x – 15<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Subtract 12x<\/p>\n<\/td>\n

      		\n

      -7 = 8x – 15<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Add 15<\/p>\n<\/td>\n

      		\n

      8 = 8x<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Divide<\/p>\n<\/td>\n

      		\n

      x = 1<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      \u00a0<\/p>\n

      Reducing Equations with Nested Parentheses<\/strong><\/p>\n

      Definition<\/u><\/strong><\/p>\n

      When parentheses are inside parentheses, work from the\u00a0innermost<\/strong>\u00a0parentheses outward.<\/p>\n

      Example:<\/strong>\u00a0Solve 2[3x – (x + 1)] = 10<\/p>\n

      \n\n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      2[3x – (x + 1)] = 10<\/p>\n<\/td>\n<\/tr>\n

      \n

      Innermost: (x+1)<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\u00a0<\/td>\n<\/tr>\n
      \n

      Remove inner: 3x – x – 1<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      2[2x – 1] = 10<\/p>\n<\/td>\n<\/tr>\n

      \n

      Distribute<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      4x – 2 = 10<\/p>\n<\/td>\n<\/tr>\n

      \n

      Add 2<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      4x = 12<\/p>\n<\/td>\n<\/tr>\n

      \n

      Divide<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      x = 3<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Example:<\/strong>\u00a0Solve 3{2x – [x + (x – 2)]} = 12<\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      3{2x – [x + (x – 2)]} = 12<\/p>\n<\/td>\n<\/tr>\n

      \n

      Innermost: (x-2)<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      3{2x – [x + x – 2]} = 12<\/p>\n<\/td>\n<\/tr>\n

      \n

      Simplify inside brackets<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      3{2x – [2x – 2]} = 12<\/p>\n<\/td>\n<\/tr>\n

      \n

      Remove brackets: 2x – 2x + 2<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      3{2} = 12<\/p>\n<\/td>\n<\/tr>\n

      \n

      Multiply<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      6 = 12? Wait, check carefully<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Let me redo carefully:<\/p>\n

      3{2x – [x + (x – 2)]} = 12<\/p>\n

        \n
      • Innermost: (x – 2) stays<\/li>\n
      • Next: [x + (x – 2)] = [x + x – 2] = [2x – 2]<\/li>\n
      • Now: 2x – [2x – 2] = 2x – 2x + 2 = 2<\/li>\n
      • So: 3{2} = 12 \u2192 6 = 12 (False) \u2192 No solution<\/li>\n<\/ul>\n

        Answer:<\/strong>\u00a0No solution<\/p>\n

        \u00a0<\/p>\n

        Variables in Denominator (Rational Equations)<\/strong><\/p>\n

        Definition<\/u><\/strong><\/p>\n

        When the variable appears in the denominator, multiply both sides by the denominator to eliminate it.\u00a0Always check for excluded values<\/strong>\u00a0(denominator \u2260 0).<\/p>\n

        Example 1:<\/strong>\u00a0Solve 3\/x = 2<\/p>\n

        \n\n\n\n\n\n\n\n\n
        \n

        Step<\/p>\n<\/td>\n

        		\n

        Operation<\/p>\n<\/td>\n

        		\n

        Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

        \n

        Given<\/p>\n<\/td>\n

        		\u00a0<\/td>\n\n

        3\/x = 2<\/p>\n<\/td>\n<\/tr>\n

        \n

        Multiply both sides by x<\/p>\n<\/td>\n

        		\n

        3 = 2x<\/p>\n<\/td>\n

        		\u00a0<\/td>\n<\/tr>\n
        \n

        Divide by 2<\/p>\n<\/td>\n

        		\n

        x = 3\/2 = 1.5<\/p>\n<\/td>\n

        		\u00a0<\/td>\n<\/tr>\n
        \n

        Check denominator<\/p>\n<\/td>\n

        		\n

        x = 1.5 \u2260 0 \u2713<\/p>\n<\/td>\n

        		\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        Source: Kapdec.com<\/p>\n<\/div>\n

        Example 2:<\/strong>\u00a0Solve 4\/(x – 1) = 2\/(x + 2)<\/p>\n

        \n\n\n\n\n\n\n\n\n\n\n\n
        \n

        Step<\/p>\n<\/td>\n

        		\n

        Operation<\/p>\n<\/td>\n

        		\n

        Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

        \n

        Given<\/p>\n<\/td>\n

        		\u00a0<\/td>\n\n

        4\/(x-1) = 2\/(x+2)<\/p>\n<\/td>\n<\/tr>\n

        \n

        Cross multiply<\/p>\n<\/td>\n

        		\n

        4(x + 2) = 2(x – 1)<\/p>\n<\/td>\n

        		\u00a0<\/td>\n<\/tr>\n
        \n

        Distribute<\/p>\n<\/td>\n

        		\n

        4x + 8 = 2x – 2<\/p>\n<\/td>\n

        		\u00a0<\/td>\n<\/tr>\n
        \n

        Subtract 2x<\/p>\n<\/td>\n

        		\n

        2x + 8 = -2<\/p>\n<\/td>\n

        		\u00a0<\/td>\n<\/tr>\n
        \n

        Subtract 8<\/p>\n<\/td>\n

        		\n

        2x = -10<\/p>\n<\/td>\n

        		\u00a0<\/td>\n<\/tr>\n
        \n

        Divide<\/p>\n<\/td>\n

        		\n

        x = -5<\/p>\n<\/td>\n

        		\u00a0<\/td>\n<\/tr>\n
        \n

        Check denominators<\/p>\n<\/td>\n

        		\n

        -5-1 = -6 \u2260 0; -5+2 = -3 \u2260 0 \u2713<\/p>\n<\/td>\n

        		\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        Source: Kapdec.com<\/p>\n<\/div>\n

        Warning:<\/strong>\u00a0If the solution makes any denominator zero, it is\u00a0extraneous\u00a0and must be rejected.<\/p>\n

        \u00a0<\/p>\n

        Solved Examples<\/strong><\/p>\n

        Example 1:<\/strong>\u00a0Reduce and solve: 3(2x + 1) – 2(x – 3) = 17<\/p>\n

        Solution:<\/strong><\/p>\n

          \n
        • Distribute: 6x + 3 – 2x + 6 = 17<\/li>\n
        • Combine: 4x + 9 = 17<\/li>\n
        • Subtract 9: 4x = 8<\/li>\n
        • Divide: x = 2<\/li>\n<\/ul>\n

          Answer:<\/strong>\u00a0x = 2<\/p>\n

          \u00a0<\/p>\n

          Example 2:<\/strong>\u00a0Reduce and solve: (x + 2)\/3 – (x – 1)\/2 = 1\/6<\/p>\n

          Solution:<\/strong><\/p>\n

            \n
          • LCM of 3, 2, 6 = 6<\/li>\n
          • Multiply by 6: 2(x + 2) – 3(x – 1) = 1<\/li>\n
          • Distribute: 2x + 4 – 3x + 3 = 1<\/li>\n
          • Combine: -x + 7 = 1<\/li>\n
          • Subtract 7: -x = -6<\/li>\n
          • Multiply by -1: x = 6<\/li>\n<\/ul>\n

            Answer:<\/strong>\u00a0x = 6<\/p>\n

            \u00a0<\/p>\n

            Example 3:<\/strong>\u00a0Reduce and solve: 0.2x – 0.5 = 0.3x + 0.1<\/p>\n

            Solution:<\/strong><\/p>\n

              \n
            • Multiply by 10: 2x – 5 = 3x + 1<\/li>\n
            • Subtract 2x: -5 = x + 1<\/li>\n
            • Subtract 1: -6 = x<\/li>\n<\/ul>\n

              Answer:<\/strong>\u00a0x = -6<\/p>\n

              \u00a0<\/p>\n

              Example 4 \u2013 Odd One Out:<\/strong><\/p>\n

              Examine the five equations below. Exactly one requires a different FIRST step to reduce. Identify it.<\/strong><\/p>\n

              \n\n\n\n\n\n\n\n\n\n
              \n

              Item<\/p>\n<\/td>\n

              		\n

              Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

              \n

              A<\/p>\n<\/td>\n

              		\n

              3(x + 2) = 2(x – 1)<\/p>\n<\/td>\n<\/tr>\n

              \n

              B<\/p>\n<\/td>\n

              		\n

              x\/2 + x\/3 = 5<\/p>\n<\/td>\n<\/tr>\n

              \n

              C<\/p>\n<\/td>\n

              		\n

              0.5x + 0.25 = 1.5<\/p>\n<\/td>\n<\/tr>\n

              \n

              D<\/p>\n<\/td>\n

              		\n

              (x + 1)\/4 = (x – 2)\/3<\/p>\n<\/td>\n<\/tr>\n

              \n

              E<\/p>\n<\/td>\n

              		\n

              2x + 5 = 3x – 2<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              Source: Kapdec.com<\/p>\n<\/div>\n

              Solution (First step analysis):<\/strong><\/p>\n

              \n\n\n\n\n\n\n\n\n\n
              \n

              Item<\/p>\n<\/td>\n

              		\n

              First Step<\/p>\n<\/td>\n

              		\n

              Type<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

              \n

              A<\/p>\n<\/td>\n

              		\n

              Distribute parentheses<\/p>\n<\/td>\n

              		\n

              Remove parentheses<\/p>\n<\/td>\n<\/tr>\n

              \n

              B<\/p>\n<\/td>\n

              		\n

              Multiply by LCM (6)<\/p>\n<\/td>\n

              		\n

              Clear fractions<\/p>\n<\/td>\n<\/tr>\n

              \n

              C<\/p>\n<\/td>\n

              		\n

              Multiply by 100 (or 100? Multiply by 100 to clear decimals? 0.5x+0.25=1.5 \u2192 \u00d7100 gives 50x+25=150, or \u00d710 gives 5x+2.5=15, still decimal. Better: \u00d710 then \u00d710 again. Actually \u00d710: 5x+2.5=15 (still decimal) \u2192 \u00d710 again: 50x+25=150. So \u00d7100)<\/p>\n<\/td>\n

              		\n

              Clear decimals<\/p>\n<\/td>\n<\/tr>\n

              \n

              D<\/p>\n<\/td>\n

              		\n

              Cross multiply<\/p>\n<\/td>\n

              		\n

              Clear fractions<\/p>\n<\/td>\n<\/tr>\n

              \n

              E<\/p>\n<\/td>\n

              		\n

              Already in simple form<\/p>\n<\/td>\n

              		\n

              None needed (or variable on both sides)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              Source: Kapdec.com<\/p>\n<\/div>\n

              Items A, B, C, D all require a reduction step. Item E is already in standard form (variables on both sides) and can be solved directly without reduction.<\/p>\n

              Three reasons why Item E is the odd one out:<\/strong><\/p>\n

              (A) Reduction need:<\/strong>\u00a0Items A, B, C, D require some form of reduction (removing parentheses, clearing fractions\/decimals). Item E requires no reduction \u2013 it is already in standard linear form 2x + 5 = 3x – 2.<\/p>\n

              (B) Complexity:<\/strong>\u00a0Items A-D have parentheses, fractions, or decimals. Item E has only integers and no grouping symbols.<\/p>\n

              (C) First operation:<\/strong>\u00a0For Items A-D, the first operation is either distribution or multiplication to clear fractions. For Item E, the first operation is subtracting a variable term from both sides.<\/p>\n

              Conclusion:<\/strong>\u00a0Item E is the odd one out.<\/p>\n

              \u00a0<\/p>\n

              <\/p>\n