https://file.aiquickdraw.com/mathbot/file/1726522905131q4uswe7o.jpg

I can see five math problems from the image. I'll walk through how to approach each one.

### 1. Solve the inequality:
\[
|x^2 - \frac{13}{2}| < \frac{5}{2}
\]

We need to solve this inequality by first breaking it into two conditions based on the absolute value function:
\[
-\frac{5}{2} < x^2 - \frac{13}{2} < \frac{5}{2}
\]

This can be broken down into two inequalities:
\[
x^2 - \frac{13}{2} < \frac{5}{2}
\]
and
\[
x^2 - \frac{13}{2} > -\frac{5}{2}
\]

### 2. Solve the inequality:
\[
|x - 5| \geq 2|x - 1|
\]

We again break this inequality into two cases based on the absolute value function. We'll consider the two situations where the absolute values can be simplified into different regions.

### 3. Solve the inequality:
\[
\left|\frac{1}{x - 1} + 2\right| \leq 1
\]

This inequality deals with a fraction inside the absolute value. We will first isolate the fraction, analyze when it's positive or negative, and then solve the resulting inequalities.

### 4. Solve the inequality:
\[
\left|\frac{1}{x} - 3\right| > 6
\]

We'll split this into two inequalities, one for the positive case and one for the negative case, and then solve them.

### 5. Measurement accuracy question:
A cylinder (graduated cylinder) has a radius of 4 cm. If we measure 0.5 cm³ of water with an error of no more than 1%, what is the accuracy of the height measurement of the water in the cylinder?

The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
Given that \( r = 4 \, \text{cm} \) and \( V = 0.5 \, \text{cm}^3 \), we can solve for \( h \), the height of the water. After finding \( h \), we will determine the 1% error in height measurement.

---

I'll solve each one step-by-step. Would you like to start with one particular question, or should I go through each of them in sequence? Let me know if you'd like detailed solutions or clarifications.

Here are 5 related questions to think about:
1. How do you handle absolute value inequalities?
2. How does changing the radius affect the volume in question 5?
3. What happens when \( x \) approaches 1 in question 3?
4. Can we express the inequalities from question 2 graphically?
5. What methods are best for solving quadratic inequalities?

**Tip**: Always check the solution set after solving inequalities, as some values might not be valid in the original equation.

Solve the inequalities and find the accuracy of the height measurement of water in a graduated cylinder with radius 4 cm and a 1% error.

Learn how to solve absolute value inequalities and calculate the accuracy of height measurement in a cylinder with radius 4 cm and a 1% error. This math problem covers key algebraic and geometric concepts suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation