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Solve the following inequality a) (2x-3)/5 >= x/2 - 1 b) (4(x+1)) < 2x + 3 c) −2(m−3) < 5(m+1) − 12

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Answers

3 parts
a

x≤4x \le 4

b

x<−12x < -\frac{1}{2}

c

m>137m > \frac{13}{7}

Step-by-step solution

13 steps

Let's solve these inequalities one by one.

Part a· 4 steps

Solve the inequality 2x−35≥x2−1\frac{2x-3}{5} \ge \frac{x}{2} - 1

  1. Find a common denominator for all terms.

    The least common multiple of 5 and 2 is 10. Multiply every term by 10 to eliminate the denominators.

    10(2x−35)≥10(x2)−10(1)10 \left(\frac{2x-3}{5}\right) \ge 10 \left(\frac{x}{2}\right) - 10(1) 2(2x−3)≥5x−102(2x-3) \ge 5x - 10

  2. Distribute and simplify both sides of the inequality.

    4x−6≥5x−104x - 6 \ge 5x - 10

  3. Gather xx terms on one side and constant terms on the other side.

    Subtract 4x4x from both sides.

    −6≥5x−4x−10-6 \ge 5x - 4x - 10 −6≥x−10-6 \ge x - 10

  4. Add 10 to both sides to isolate xx.

    −6+10≥x-6 + 10 \ge x 4≥x4 \ge x

    This can also be written as x≤4x \le 4.

    x≤4\boxed{x \le 4}

Part b

x<−12x < -\frac{1}{2}

Solve the inequality 4(x+1)<2x+34(x+1) < 2x + 3

  1. Distribute the 4 on the left side of the inequality.

    4x+4<2x+34x + 4 < 2x + 3

  2. Gather xx terms on one side and constant terms on the other.

    Subtract 2x2x from both sides.

    4x−2x+4<34x - 2x + 4 < 3 2x+4<32x + 4 < 3

  3. Subtract 4 from both sides to isolate the xx term.

    2x<3−42x < 3 - 4 2x<−12x < -1

  4. Divide by 2 to solve for xx.

    x<−12x < -\frac{1}{2} x<−12\boxed{x < -\frac{1}{2}}

Part c

m>137m > \frac{13}{7}

Solve the inequality −2(m−3)<5(m+1)−12-2(m-3) < 5(m+1) - 12

  1. Distribute the numbers on both sides of the inequality.

    −2m+6<5m+5−12-2m + 6 < 5m + 5 - 12

  2. Simplify the right side by combining constant terms.

    −2m+6<5m−7-2m + 6 < 5m - 7

  3. Gather mm terms on one side and constant terms on the other.

    Add 2m2m to both sides.

    6<5m+2m−76 < 5m + 2m - 7 6<7m−76 < 7m - 7

  4. Add 7 to both sides to isolate the mm term.

    6+7<7m6 + 7 < 7m 13<7m13 < 7m

  5. Divide by 7 to solve for mm.

    137<m\frac{13}{7} < m

    This can also be written as m>137m > \frac{13}{7}.

    m>137\boxed{m > \frac{13}{7}}
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