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Factorize completely: (h2 - k2) - p(h + k). Solve: 2x - 3 <= 4x + 5 < 3x + 7.

Mathematics

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D

Step-by-step solution

7 steps

Here are the solutions to questions 4 and 5.

Part 4· 4 steps

Factorize completely (h2−k2)−p(h+k)(h^2 - k^2) - p(h + k)

  1. Factor the difference of squares term.

    The expression h2−k2h^2 - k^2 is a difference of squares, which factors as (h−k)(h+k)(h - k)(h + k).

  2. Substitute the factored term back into the expression.

    (h−k)(h+k)−p(h+k)(h - k)(h + k) - p(h + k)

  3. Factor out the common term (h+k)(h + k).

    (h+k)[(h−k)−p](h + k)[(h - k) - p]

  4. Simplify the expression inside the brackets.

    (h+k)(h−k−p)(h + k)(h - k - p)

    The correct option is D.

Part 5· 3 steps

Solve 2x−3≤4x+5<3x+72x - 3 \le 4x + 5 < 3x + 7.

This compound inequality can be split into two separate inequalities:

  1. 2x−3≤4x+52x - 3 \le 4x + 5
  2. 4x+5<3x+74x + 5 < 3x + 7

  1. Solve the first inequality, 2x−3≤4x+52x - 3 \le 4x + 5.

    Subtract 2x2x from both sides:

    −3≤2x+5-3 \le 2x + 5

    Subtract 5 from both sides:

    −3−5≤2x-3 - 5 \le 2x −8≤2x-8 \le 2x

    Divide by 2:

    −82≤x\frac{-8}{2} \le x −4≤x-4 \le x

  2. Solve the second inequality, 4x+5<3x+74x + 5 < 3x + 7.

    Subtract 3x3x from both sides:

    x+5<7x + 5 < 7

    Subtract 5 from both sides:

    x<7−5x < 7 - 5 x<2x < 2

  3. Combine the solutions from Step 1 and Step 2.

    We have x≥−4x \ge -4 and x<2x < 2. Combining these gives:

    −4≤x<2-4 \le x < 2

    The correct option is D.

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