https://file.aiquickdraw.com/mathbot/file/1725417756999rzvvlop7.jpg

Let's solve the problems from the image step-by-step:

### 1. Solve for \( x \): 
\[
\frac{2x - 1}{3} = 5
\]

Multiply both sides by 3 to eliminate the denominator:
\[
2x - 1 = 15
\]

Add 1 to both sides:
\[
2x = 16
\]

Divide by 2:
\[
x = 8
\]

### 2. Solve for \( x \):
\[
\frac{x + 2}{5} = \frac{x + 4}{2}
\]

Cross-multiply:
\[
2(x + 2) = 5(x + 4)
\]

Expand and simplify:
\[
2x + 4 = 5x + 20
\]

Rearrange:
\[
4 - 20 = 5x - 2x
\]

\[
-16 = 3x
\]

Divide by 3:
\[
x = -\frac{16}{3}
\]

### 3. Solve for \( x \):
\[
\frac{x+1}{x-3} = \frac{3}{2}
\]

Cross-multiply:
\[
2(x + 1) = 3(x - 3)
\]

Expand and simplify:
\[
2x + 2 = 3x - 9
\]

Rearrange:
\[
2 + 9 = 3x - 2x
\]

\[
11 = x
\]

### 4. Solve for \( x \):
\[
\frac{x-1}{2x+3} = \frac{x+3}{x-2}
\]

Cross-multiply:
\[
(x - 1)(x - 2) = (x + 3)(2x + 3)
\]

Expand both sides:
\[
x^2 - 2x - x + 2 = 2x^2 + 6x + 9
\]

Simplify:
\[
x^2 - 3x + 2 = 2x^2 + 6x + 9
\]

Rearrange:
\[
0 = x^2 + 9x + 7
\]

Solve using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[
x = \frac{-9 \pm \sqrt{81 - 28}}{2} = \frac{-9 \pm \sqrt{53}}{2}
\]

### 5. Solve for \( x \):
\[
\frac{x^2 - x - 30}{x - 1} \geq 0
\]

Factorize the numerator:
\[
\frac{(x - 6)(x + 5)}{x - 1} \geq 0
\]

Find the critical points: \(x = 6, x = -5, x = 1\).

Test intervals to determine where the expression is non-negative.

Would you like more details on any of these steps, or have any specific questions?

### Related Questions
1. How do you solve quadratic inequalities step-by-step?
2. What are some methods to solve rational equations?
3. How can you solve systems of linear equations?
4. What are some common techniques for factoring polynomials?
5. How do you determine the domain and range of a function?

### Tip
Always check for any restrictions on the domain of the solution, especially when dealing with rational expressions.

Lesson 6 Supplementary Exercises: Solve for x in various algebraic equations and inequalities.

Explore detailed solutions to Lesson 6 Supplementary Exercises, including solving linear equations, rational equations, and quadratic inequalities. Suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation