Math Problem Statement

Solve equations, inequalities, and systems of equations as presented in the image.

Solution

I'll solve each of the problems in your image. Here's the breakdown:


Zad. 8 (Solve the equations)

(a)

3x52+3=3x+56\frac{3x - 5}{2} + 3 = \frac{3x + 5}{6}

  1. Eliminate the fractions by multiplying through by the least common denominator (LCD = 6): 63x52+63=63x+566 \cdot \frac{3x - 5}{2} + 6 \cdot 3 = 6 \cdot \frac{3x + 5}{6} 3(3x5)+18=3x+53(3x - 5) + 18 = 3x + 5

  2. Expand and simplify: 9x15+18=3x+59x - 15 + 18 = 3x + 5 9x+3=3x+59x + 3 = 3x + 5

  3. Solve for xx: 9x3x=539x - 3x = 5 - 3 6x=2x=136x = 2 \quad \Rightarrow \quad x = \frac{1}{3}

(b)

x+14=0|x + 1| - 4 = 0

  1. Rearrange: x+1=4|x + 1| = 4

  2. Solve the absolute value equation: x+1=4orx+1=4x + 1 = 4 \quad \text{or} \quad x + 1 = -4 x=3orx=5x = 3 \quad \text{or} \quad x = -5


Zad. 9 (Solve the inequalities)

(a)

23(x1)4122x+18\frac{2 - 3(x - 1)}{4} \geq \frac{1}{2} \cdot \frac{2x + 1}{8}

  1. Simplify each term and multiply through by the LCD (8): 823(x1)48122x+188 \cdot \frac{2 - 3(x - 1)}{4} \geq 8 \cdot \frac{1}{2} \cdot \frac{2x + 1}{8} 2(23(x1))2x+12(2 - 3(x - 1)) \geq 2x + 1

  2. Expand: 46(x1)2x+14 - 6(x - 1) \geq 2x + 1 46x+62x+14 - 6x + 6 \geq 2x + 1

  3. Simplify: 106x2x+110 - 6x \geq 2x + 1

  4. Solve for xx: 1016x+2x10 - 1 \geq 6x + 2x 98xx989 \geq 8x \quad \Rightarrow \quad x \leq \frac{9}{8}

(b)

x7<4|x - 7| < 4

  1. Solve the absolute value inequality: 4<x7<4-4 < x - 7 < 4

  2. Add 7 to all parts: 3<x<113 < x < 11


Zad. 10 (Solve the system of equations)

4x + 7y &= 7 \\ 3x + 6y &= 3 \end{aligned}$$ 1. Simplify the second equation: $$\frac{3x + 6y}{3} = \frac{3}{3} \quad \Rightarrow \quad x + 2y = 1$$ 2. Solve the system: From $$x + 2y = 1$$, express $$x$$: $$x = 1 - 2y$$ 3. Substitute $$x = 1 - 2y$$ into $$4x + 7y = 7$$: $$4(1 - 2y) + 7y = 7$$ $$4 - 8y + 7y = 7$$ $$4 - y = 7$$ $$y = -3$$ 4. Substitute $$y = -3$$ into $$x + 2y = 1$$: $$x + 2(-3) = 1$$ $$x - 6 = 1 \quad \Rightarrow \quad x = 7$$ Solution: $$x = 7, \, y = -3$$ --- Do you want a detailed explanation of any step? ### Related Questions: 1. How do you solve absolute value inequalities generally? 2. What are strategies for solving equations with fractions? 3. How can systems of equations be solved using matrices? 4. What are the rules for combining inequalities? 5. What is the geometric interpretation of absolute value equations? **Tip:** Always double-check your solution by substituting it back into the original equation or inequality!

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Absolute Value Equations
Linear Inequalities
Systems of Linear Equations

Formulas

Eliminating fractions by multiplying by the least common denominator (LCD)
Absolute value property: |x| = a ⇔ x = a or x = -a
Inequality solving rules
Methods for solving systems of equations (substitution and elimination)

Theorems

Properties of Absolute Values
Basic Algebraic Principles
Rules for Inequalities

Suitable Grade Level

Grades 8-10