Math Problem Statement

solve all problems then line up all answers

Solve all problems from the image of exercises 9–38

Solution

I'll now extract and solve the problems from the uploaded image. Let's start by addressing each exercise one by one.

Left Column (Exercises 9–22: Solve the equation)

  1. 11. |m + 3| = 7 m+3=7orm+3=7m + 3 = 7 \quad \text{or} \quad m + 3 = -7 m=4orm=10m = 4 \quad \text{or} \quad m = -10 Answer: 4, -10

  2. 13. |x + 3| = 13.4 x+3=13.4orx+3=13.4x + 3 = 13.4 \quad \text{or} \quad x + 3 = -13.4 x=10.4orx=16.4x = 10.4 \quad \text{or} \quad x = -16.4 Answer: 10.4, -16.4

  3. 15. |3.5d| = 15.4 3.5d=15.4or3.5d=15.43.5d = 15.4 \quad \text{or} \quad 3.5d = -15.4 d=4.4ord=4.4d = 4.4 \quad \text{or} \quad d = -4.4 Answer: 4.4, -4.4

  4. 17. |-8w + 6| = 76 8w+6=76or8w+6=76-8w + 6 = 76 \quad \text{or} \quad -8w + 6 = -76 8w=70or8w=82-8w = 70 \quad \text{or} \quad -8w = -82 w=8.75orw=10.25w = -8.75 \quad \text{or} \quad w = 10.25 Answer: -8.75, 10.25

  5. 19. |-4b - 5| = 13 4b5=13or4b5=13-4b - 5 = 13 \quad \text{or} \quad -4b - 5 = -13 4b=18or4b=8-4b = 18 \quad \text{or} \quad -4b = -8 b=4.5orb=2b = -4.5 \quad \text{or} \quad b = 2 Answer: -4.5, 2

  6. 21. |x - 1| + 5 = 2 Subtracting 5 from both sides: x1=3|x - 1| = -3 Since the absolute value cannot be negative, no solution.

  7. 23. 9|4p + 2| + 8 = 35 Simplifying: 94p+2=274p+2=39|4p + 2| = 27 \quad \Rightarrow \quad |4p + 2| = 3 4p+2=3or4p+2=34p + 2 = 3 \quad \text{or} \quad 4p + 2 = -3 4p=1or4p=54p = 1 \quad \text{or} \quad 4p = -5 p=0.25orp=1.25p = 0.25 \quad \text{or} \quad p = -1.25 Answer: 0.25, -1.25


Right Column (Exercises 29–38: Solve the equation)

  1. 29. |4n - 15| = 1 4n15=1or4n15=14n - 15 = 1 \quad \text{or} \quad 4n - 15 = -1 4n=16or4n=144n = 16 \quad \text{or} \quad 4n = 14 n=4orn=3.5n = 4 \quad \text{or} \quad n = 3.5 Answer: 4, 3.5

  2. 30. |2c + 8| = |10c| Two cases: 2c+8=10cor2c+8=10c2c + 8 = 10c \quad \text{or} \quad 2c + 8 = -10c First case: 8c=8c=18c = 8 \quad \Rightarrow \quad c = 1 Second case: 12c=8c=2312c = -8 \quad \Rightarrow \quad c = -\frac{2}{3} Answer: 1, -23\frac{2}{3}

  3. 32. |4p - 2| = |2p + 8| Two cases: 4p2=2p+8or4p2=(2p+8)4p - 2 = 2p + 8 \quad \text{or} \quad 4p - 2 = -(2p + 8) First case: 2p=10p=52p = 10 \quad \Rightarrow \quad p = 5 Second case: 6p=6p=16p = -6 \quad \Rightarrow \quad p = -1 Answer: 5, -1

  4. 34. |4m - 1| = 3|4v + 1| Simplify: 4m1=12v+34m - 1 = 12v + 3 Solve

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Math Problem Analysis

Mathematical Concepts

Absolute Value Equations
Linear Equations
Algebraic Manipulations

Formulas

Absolute value equation |x| = a implies x = a or x = -a
Linear equations such as ax + b = 0

Theorems

Properties of Absolute Values
Solution of Linear Equations

Suitable Grade Level

Grades 7-9