Solve the following inequalities, starting the solution in words.

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Solve the following inequalities, starting the solution in words. Graph the inequality on the number line. (a) x – 25 > 7 (b) 5 < 2x + 15 (c) 25/x < 10 (d) x/2 + x/2 + x/3 ≥ 17/6 (e) 3x – 29 ≤ 7x + 11

The Correct Answer and Explanation is:

Let’s go through each inequality step by step and explain the solutions.

(a) x−25>7x – 25 > 7x−25>7

  1. Start by isolating xxx. To do this, add 25 to both sides: x>7+25x > 7 + 25x>7+25 x>32x > 32x>32 Solution: x>32x > 32x>32. This means that xxx must be greater than 32.
  2. Graphing: On the number line, represent an open circle at 32, and shade to the right of 32 to show all values greater than 32.

(b) 5<2x+155 < 2x + 155<2x+15

  1. Isolate the xxx-term. Subtract 15 from both sides: 5−15<2×5 – 15 < 2×5−15<2x −10<2x-10 < 2x−10<2x
  2. Solve for xxx by dividing both sides by 2: −102<x\frac{-10}{2} < x2−10​<x −5<x-5 < x−5<x Solution: x>−5x > -5x>−5. This means that xxx is greater than -5.
  3. Graphing: On the number line, represent an open circle at -5, and shade to the right of -5.

(c) 25x<10\frac{25}{x} < 10×25​<10

  1. Isolate xxx. Multiply both sides by xxx (but remember, this changes the direction of the inequality if xxx is negative): 25<10×25 < 10×25<10x
  2. Solve for xxx by dividing both sides by 10: 2510<x\frac{25}{10} < x1025​<x 2.5<x2.5 < x2.5<x Solution: x>2.5x > 2.5x>2.5. This means that xxx must be greater than 2.5.
  3. Graphing: On the number line, represent an open circle at 2.5, and shade to the right of 2.5.

(d) x2+x2+x3≥176\frac{x}{2} + \frac{x}{2} + \frac{x}{3} \geq \frac{17}{6}2x​+2x​+3x​≥617​

  1. Combine like terms. The first two terms can be added together: x2+x2=2×2=x\frac{x}{2} + \frac{x}{2} = \frac{2x}{2} = x2x​+2x​=22x​=x Now the inequality becomes: x+x3≥176x + \frac{x}{3} \geq \frac{17}{6}x+3x​≥617​
  2. Get a common denominator for the terms on the left-hand side. The common denominator between 1 and 3 is 3: 3×3+x3=4×3\frac{3x}{3} + \frac{x}{3} = \frac{4x}{3}33x​+3x​=34x​ So the inequality becomes: 4×3≥176\frac{4x}{3} \geq \frac{17}{6}34x​≥617​
  3. Clear the fraction by multiplying both sides by 6 (the least common denominator of 3 and 6): 6×4×3≥6×1766 \times \frac{4x}{3} \geq 6 \times \frac{17}{6}6×34x​≥6×617​ 8x≥178x \geq 178x≥17
  4. Solve for xxx by dividing both sides by 8: x≥178x \geq \frac{17}{8}x≥817​ x≥2.125x \geq 2.125x≥2.125 Solution: x≥2.125x \geq 2.125x≥2.125. This means xxx is greater than or equal to 2.125.
  5. Graphing: On the number line, represent a closed circle at 2.125, and shade to the right of it.

(e) 3x−29≤7x+113x – 29 \leq 7x + 113x−29≤7x+11

  1. Move all terms involving xxx to one side. Subtract 3x3x3x from both sides: −29≤4x+11-29 \leq 4x + 11−29≤4x+11
  2. Move the constants to the other side by subtracting 11 from both sides: −29−11≤4x-29 – 11 \leq 4x−29−11≤4x −40≤4x-40 \leq 4x−40≤4x
  3. Solve for xxx by dividing both sides by 4: −404≤x\frac{-40}{4} \leq x4−40​≤x −10≤x-10 \leq x−10≤x Solution: x≥−10x \geq -10x≥−10. This means xxx is greater than or equal to -10.
  4. Graphing: On the number line, represent a closed circle at -10, and shade to the right of it.

Summary of Solutions:

  • (a) x>32x > 32x>32
  • (b) x>−5x > -5x>−5
  • (c) x>2.5x > 2.5x>2.5
  • (d) x≥2.125x \geq 2.125x≥2.125
  • (e) x≥−10x \geq -10x≥−10

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