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Given A, B, and C matrices, evaluate A-B+C.

Mathematics

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Answer

b

−2213<x≤813-\frac{22}{13} < x \le \frac{8}{13}

Step-by-step solution

4 steps

Question 1(a)

Step 1: Separate the compound inequality into two inequalities.
The given inequality is −4<2x−13(1−12x)≤1-4 < 2x - \frac{1}{3}(1 - \frac{1}{2}x) \le 1.
This can be split into:

Part a

−4<2x−13(1−12x)-4 < 2x - \frac{1}{3}(1 - \frac{1}{2}x)

Part b

−2213<x≤813-\frac{22}{13} < x \le \frac{8}{13}

2x−13(1−12x)≤12x - \frac{1}{3}(1 - \frac{1}{2}x) \le 1

  1. Solve the first inequality a).

    −4<2x−13+16x-4 < 2x - \frac{1}{3} + \frac{1}{6}x −4<(2+16)x−13-4 < \left(2 + \frac{1}{6}\right)x - \frac{1}{3} −4<136x−13-4 < \frac{13}{6}x - \frac{1}{3}

    Add 13\frac{1}{3} to both sides:

    −4+13<136x-4 + \frac{1}{3} < \frac{13}{6}x −123+13<136x-\frac{12}{3} + \frac{1}{3} < \frac{13}{6}x −113<136x-\frac{11}{3} < \frac{13}{6}x

    Multiply by 613\frac{6}{13}:

    −113⋅613<x-\frac{11}{3} \cdot \frac{6}{13} < x −2213<x-\frac{22}{13} < x
  2. Solve the second inequality b).

    2x−13(1−12x)≤12x - \frac{1}{3}(1 - \frac{1}{2}x) \le 1 2x−13+16x≤12x - \frac{1}{3} + \frac{1}{6}x \le 1 (2+16)x−13≤1\left(2 + \frac{1}{6}\right)x - \frac{1}{3} \le 1 136x−13≤1\frac{13}{6}x - \frac{1}{3} \le 1

    Add 13\frac{1}{3} to both sides:

    136x≤1+13\frac{13}{6}x \le 1 + \frac{1}{3} 136x≤33+13\frac{13}{6}x \le \frac{3}{3} + \frac{1}{3} 136x≤43\frac{13}{6}x \le \frac{4}{3}

    Multiply by 613\frac{6}{13}:

    x≤43⋅613x \le \frac{4}{3} \cdot \frac{6}{13} x≤813x \le \frac{8}{13}
  3. Combine the solutions from Step 2 and Step 3.

    We have x>−2213x > -\frac{22}{13} and x≤813x \le \frac{8}{13}.
    The solution set is −2213<x≤813-\frac{22}{13} < x \le \frac{8}{13}.

  4. Illustrate the solution on a number line.

    The interval is (−2213,813](-\frac{22}{13}, \frac{8}{13}].
    On the number line, place an open circle at −2213-\frac{22}{13} (approximately −1.69-1.69) and a closed circle at 813\frac{8}{13} (approximately 0.62). Shade the region between these two points.

    <--------------------------------------------------------------------->
          -2          -22/13          -1           0          8/13         1
    <-----o----------------------------------------------------•------------>
    

    -\frac{22}{13} < x \le \frac{8}{13}

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