https://r2.erweima.ai/mathbot/image/1726248062022p910xv02.jpg

Let's tackle the inequalities for the problems 413 and 414 from your image step-by-step.

### Problem 413: 

\[
\frac{3x + 1}{2x - 4} \geq \frac{2}{x - 5}
\]

First, let's move all terms to one side:

\[
\frac{3x + 1}{2x - 4} - \frac{2}{x - 5} \geq 0
\]

To solve this inequality, let's get a common denominator. The common denominator is \((2x - 4)(x - 5)\):

\[
\frac{(3x + 1)(x - 5) - 2(2x - 4)}{(2x - 4)(x - 5)} \geq 0
\]

Expanding the numerators:

\[
(3x + 1)(x - 5) = 3x^2 - 15x + x - 5 = 3x^2 - 14x - 5
\]
\[
2(2x - 4) = 4x - 8
\]

Now subtract these:

\[
3x^2 - 14x - 5 - (4x - 8) = 3x^2 - 14x - 5 - 4x + 8 = 3x^2 - 18x + 3
\]

The inequality becomes:

\[
\frac{3x^2 - 18x + 3}{(2x - 4)(x - 5)} \geq 0
\]

Now, let's solve this by finding the critical points. The numerator \(3x^2 - 18x + 3\) can be factored as:

\[
3(x^2 - 6x + 1)
\]

To find the roots of \(x^2 - 6x + 1 = 0\), we apply the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(1)}}{2(1)} = \frac{6 \pm \sqrt{36 - 4}}{2} = \frac{6 \pm \sqrt{32}}{2} = \frac{6 \pm 4\sqrt{2}}{2} = 3 \pm 2\sqrt{2}
\]

So the roots are \(x = 3 + 2\sqrt{2}\) and \(x = 3 - 2\sqrt{2}\).

For the denominator:

\[
(2x - 4)(x - 5) = 2(x - 2)(x - 5)
\]

The roots of the denominator are \(x = 2\) and \(x = 5\).

Now we can draw a sign chart considering the critical points:

- From the numerator: \(x = 3 + 2\sqrt{2}\), \(x = 3 - 2\sqrt{2}\)
- From the denominator: \(x = 2\), \(x = 5\)

Check the intervals around these points, keeping in mind the sign changes and exclusions due to denominators.

---

### Problem 414:

\[
\frac{7(x + 2)}{-x + 3} + \frac{5x}{3 - 2x} + \frac{2(x^2 + x - 17)}{9 - 2x^2 - 3x} \leq 0
\]

This is a more complicated expression, so we need to simplify each fraction and analyze the denominator carefully. Let's break it down:

1. The first term: \(\frac{7(x + 2)}{-x + 3}\)
   - This simplifies directly. The critical point is \(x = 3\), where the denominator becomes zero.

2. The second term: \(\frac{5x}{3 - 2x}\)
   - The critical point here is \(x = \frac{3}{2}\).

3. The third term: \(\frac{2(x^2 + x - 17)}{9 - 2x^2 - 3x}\)
   - Factor the denominator \(9 - 2x^2 - 3x\), which will give the critical points.

Next, we need to factor these terms, combine them into a single inequality with a common denominator, and solve it by finding critical points. Once again, we will use a sign chart to determine where the expression is less than or equal to zero.

Would you like me to continue solving these in more detail, or should I focus on one part of the equations specifically?

### Follow-Up Questions:
1. Would you like help with factoring the expressions in problem 414?
2. Should I show the steps for constructing the sign chart for both inequalities?
3. Would you prefer a graphical or algebraic approach to visualize these inequalities?
4. Would you like further clarification on the quadratic formula used in problem 413?
5. Are there any specific steps in the common denominator process you'd like me to explain?

**Tip:** When solving rational inequalities, always be careful with the points where the denominator equals zero—these must be excluded from the solution set!

Detailed step-by-step solution of rational inequalities problems 413 and 414. Learn to solve algebraic expressions by factoring, using the quadratic formula, and applying the sign chart method. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation