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Solve the following inequalities: 1. 8m − 3(2m − 4) > 4 2. 2 - 1/2x < 5 3. 2 - (x-1)/3 >= 8 4. 1/2(x-1) - 1/3(x+5) < 1 5. (x-5)/4 - (x-3)/2 > 0

Mathematics

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Step-by-step solution

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Here are the solutions to the inequalities:

1.

  1. Distribute the −3-3 on the left side.

    8m−3(2m−4)>48m - 3(2m - 4) > 4 8m−6m+12>48m - 6m + 12 > 4

  2. Combine like terms.

    2m+12>42m + 12 > 4

  3. Subtract 12 from both sides.

    2m>4−122m > 4 - 12 2m>−82m > -8

  4. Divide by 2.

    m>−82m > \frac{-8}{2} m>−4\boxed{m > -4}

    2.

  5. Subtract 2 from both sides.

    2−12x<52 - \frac{1}{2}x < 5 −12x<5−2-\frac{1}{2}x < 5 - 2 −12x<3-\frac{1}{2}x < 3

  6. Multiply both sides by −2-2.

    Remember to reverse the inequality sign.

    x>3×(−2)x > 3 \times (-2) x>−6\boxed{x > -6}

    3.

  7. Subtract 2 from both sides.

    2−(x−1)3≥82 - \frac{(x-1)}{3} \ge 8 −(x−1)3≥8−2-\frac{(x-1)}{3} \ge 8 - 2 −(x−1)3≥6-\frac{(x-1)}{3} \ge 6

  8. Multiply both sides by −3-3.

    Remember to reverse the inequality sign.

    (x−1)≤6×(−3)(x-1) \le 6 \times (-3) x−1≤−18x-1 \le -18

  9. Add 1 to both sides.

    x≤−18+1x \le -18 + 1 x≤−17\boxed{x \le -17}

    4.

  10. Multiply the entire inequality by the least common multiple of the denominators (2 and 3), which is 6.

    12(x−1)−13(x+5)<1\frac{1}{2}(x-1) - \frac{1}{3}(x+5) < 1 6×12(x−1)−6×13(x+5)<6×16 \times \frac{1}{2}(x-1) - 6 \times \frac{1}{3}(x+5) < 6 \times 1 3(x−1)−2(x+5)<63(x-1) - 2(x+5) < 6

  11. Distribute the 3 and the −2-2.

    3x−3−2x−10<63x - 3 - 2x - 10 < 6

  12. Combine like terms.

    (3x−2x)+(−3−10)<6(3x - 2x) + (-3 - 10) < 6 x−13<6x - 13 < 6

  13. Add 13 to both sides.

    x<6+13x < 6 + 13 x<19\boxed{x < 19}

    5.

  14. Multiply the entire inequality by the least common multiple of the denominators (3 and 6), which is 6.

    23(y−3)≤56(y+6)\frac{2}{3}(y-3) \le \frac{5}{6}(y+6) 6×23(y−3)≤6×56(y+6)6 \times \frac{2}{3}(y-3) \le 6 \times \frac{5}{6}(y+6) 2×2(y−3)≤5(y+6)2 \times 2(y-3) \le 5(y+6) 4(y−3)≤5(y+6)4(y-3) \le 5(y+6)

  15. Distribute the 4 and the 5.

    4y−12≤5y+304y - 12 \le 5y + 30

  16. Subtract $4y$ from both sides.

    −12≤5y−4y+30-12 \le 5y - 4y + 30 −12≤y+30-12 \le y + 30

  17. Subtract 30 from both sides.

    −12−30≤y-12 - 30 \le y −42≤y-42 \le y

    This can also be written as:

    y≥−42\boxed{y \ge -42}
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