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Simplify the following: (2)/(x+2)-(4)/(2x+1)

Simplify the following:\newline2x+2−42x+1\frac{2}{x+2}-\frac{4}{2x+1}

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Full solution

Q. Simplify the following:\newline2x+2−42x+1\frac{2}{x+2}-\frac{4}{2x+1}
  1. Identify Terms and Denominators: Identify the terms and their denominators. We have two fractions with different denominators: 2x+2\frac{2}{x+2} and 42x+1\frac{4}{2x+1}.
  2. Find Common Denominator: Find a common denominator for the two fractions. The common denominator will be the product of the two distinct denominators, which is x+\(2)(22x+11)\.
  3. Rewrite with Common Denominator: Rewrite each fraction with the common denominator. The first fraction becomes (2)(2x+1)/[(x+2)(2x+1)](2)(2x+1)/[(x+2)(2x+1)], and the second fraction becomes (4)(x+2)/[(x+2)(2x+1)](4)(x+2)/[(x+2)(2x+1)].
  4. Expand Numerators: Expand the numerators of both fractions. For the first fraction, (2)(2x+1)(2)(2x+1) becomes 4x+24x+2. For the second fraction, (4)(x+2)(4)(x+2) becomes 4x+84x+8.
  5. Combine Fractions: Combine the two fractions by subtracting the second fraction from the first, using the common denominator. This gives us (4x+2−(4x+8))/[(x+2)(2x+1)](4x+2 - (4x+8))/[(x+2)(2x+1)].
  6. Simplify Numerator: Simplify the numerator by combining like terms. Subtracting 4x+84x+8 from 4x+24x+2 gives us −6-6.
  7. Write Final Expression: Write the simplified expression. The final simplified expression is (−6)/[(x+2)(2x+1)](-6)/[(x+2)(2x+1)].

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