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Solve and find the solution. a) −6 > 2x > 2 b) −12 < 3x < 6 c) -1 <= x+2 <= 7 d) −5 < 8x −1 <= 7 e) -3 <= 3x + 3 <= 9 f) -8 <= x -1 <= 5 g) 2y+3 < 11 h) 2x −5 > 4x +7

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8 parts
a

No solution\text{No solution}

b

−4<x<2-4 < x < 2

c

−3≤x≤5-3 \le x \le 5

d

−4<x<8-4 < x < 8

e

−2≤x≤2-2 \le x \le 2

f

−1≤x≤6-1 \le x \le 6

g

y<4y < 4

h

−6>x-6 > x

Step-by-step solution

21 steps

Here are the solutions for the given inequalities:

Part a· 3 steps

Solve −6>2x>2-6 > 2x > 2

  1. Split the compound inequality into two separate inequalities.

    −6>2xand2x>2-6 > 2x \quad \text{and} \quad 2x > 2

  2. Solve each inequality for xx.

    For the first inequality:

    −62>2x2\frac{-6}{2} > \frac{2x}{2} −3>x-3 > x

    For the second inequality:

    2x2>22\frac{2x}{2} > \frac{2}{2} x>1x > 1

  3. Combine the solutions.

    We need x<−3x < -3 and x>1x > 1. There is no value of xx that can be simultaneously less than −3-3 and greater than 1.
    The solution set is empty.

    No solution\boxed{\text{No solution}}

Part b

−4<x<2-4 < x < 2

Solve −12<3x<6-12 < 3x < 6

  1. Divide all parts of the inequality by 3.

    −123<3x3<63\frac{-12}{3} < \frac{3x}{3} < \frac{6}{3}

  2. Simplify the inequality.

    −4<x<2\boxed{-4 < x < 2}

Part c

−3≤x≤5-3 \le x \le 5

Solve −1≤x+2≤7-1 \le x+2 \le 7

  1. Subtract 2 from all parts of the inequality.

    −1−2≤x+2−2≤7−2-1 - 2 \le x+2 - 2 \le 7 - 2

  2. Simplify the inequality.

    −3≤x≤5\boxed{-3 \le x \le 5}

Part d

−4<x<8-4 < x < 8

Solve −5<x−1<7-5 < x-1 < 7

  1. Add 1 to all parts of the inequality.

    −5+1<x−1+1<7+1-5 + 1 < x-1 + 1 < 7 + 1

  2. Simplify the inequality.

    −4<x<8\boxed{-4 < x < 8}

Part e

−2≤x≤2-2 \le x \le 2

Solve −3≤3x+3≤9-3 \le 3x+3 \le 9

  1. Subtract 3 from all parts of the inequality.

    −3−3≤3x+3−3≤9−3-3 - 3 \le 3x+3 - 3 \le 9 - 3 −6≤3x≤6-6 \le 3x \le 6

  2. Divide all parts of the inequality by 3.

    −63≤3x3≤63\frac{-6}{3} \le \frac{3x}{3} \le \frac{6}{3}

  3. Simplify the inequality.

    −2≤x≤2\boxed{-2 \le x \le 2}

Part f

−1≤x≤6-1 \le x \le 6

Solve −2≤x−1≤5-2 \le x-1 \le 5

  1. Add 1 to all parts of the inequality.

    −2+1≤x−1+1≤5+1-2 + 1 \le x-1 + 1 \le 5 + 1

  2. Simplify the inequality.

    −1≤x≤6\boxed{-1 \le x \le 6}

Part g

y<4y < 4

Solve 2y+3<112y+3 < 11

  1. Subtract 3 from both sides of the inequality.

    2y+3−3<11−32y+3 - 3 < 11 - 3 2y<82y < 8

  2. Divide both sides by 2.

    2y2<82\frac{2y}{2} < \frac{8}{2}

  3. Simplify the inequality.

    y<4\boxed{y < 4}

Part h

−6>x-6 > x

Solve 2x−5>4x+72x-5 > 4x+7

  1. Subtract $2x$ from both sides of the inequality.

    2x−5−2x>4x+7−2x2x-5 - 2x > 4x+7 - 2x −5>2x+7-5 > 2x+7

  2. Subtract 7 from both sides of the inequality.

    −5−7>2x+7−7-5 - 7 > 2x+7 - 7 −12>2x-12 > 2x

  3. Divide both sides by 2.

    −122>2x2\frac{-12}{2} > \frac{2x}{2}

  4. Simplify the inequality.

    −6>x\boxed{-6 > x}
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