Solve To Determine The Solution Set Of The Inequality ∣ X + 3 ∣ \textless 5 |x+3|\ \textless \ 5 ∣ X + 3∣ \textless 5 .1. Rewrite For Two Solutions: ( X + 3 ) \textless 5 (x+3)\ \textless \ 5 ( X + 3 ) \textless 5 And $-(x+3)\ \textless \ 5$2. Solve The First Inequality: $x \ \textless \

Introduction

Absolute value inequalities are a type of mathematical expression that involves the absolute value of a variable or expression. In this article, we will focus on solving absolute value inequalities of the form $|x-a| < b$, where $a$ and $b$ are constants. We will use the given inequality $|x+3| < 5$ as an example to demonstrate the steps involved in solving absolute value inequalities.

Rewriting the Inequality

To solve the absolute value inequality $|x+3| < 5$, we need to rewrite it as two separate inequalities. This is done by creating two equations:

1. $(x+3) < 5$
2. $-(x+3) < 5$

These two equations are derived by removing the absolute value sign and considering the positive and negative cases separately.

Solving the First Inequality

Let's start by solving the first inequality: $(x+3) < 5$. To do this, we need to isolate the variable $x$.

Step 1: Subtract 3 from both sides

Subtracting 3 from both sides of the inequality gives us:

$x + 3 - 3 < 5 - 3$

This simplifies to:

$x < 2$

Step 2: Write the solution in interval notation

The solution to the inequality $x < 2$ can be written in interval notation as:

$(-\infty, 2)$

This means that $x$ can take on any value less than 2.

Solving the Second Inequality

Now, let's solve the second inequality: $-(x+3) < 5$. To do this, we need to isolate the variable $x$.

Step 1: Distribute the negative sign

Distributing the negative sign to the terms inside the parentheses gives us:

$-x - 3 < 5$

Step 2: Add 3 to both sides

Adding 3 to both sides of the inequality gives us:

$-x - 3 + 3 < 5 + 3$

This simplifies to:

$-x < 8$

Step 3: Multiply both sides by -1

Multiplying both sides of the inequality by -1 gives us:

$-(-x) > -8$

This simplifies to:

$x > -8$

Step 4: Write the solution in interval notation

The solution to the inequality $x > -8$ can be written in interval notation as:

$(-8, \infty)$

This means that $x$ can take on any value greater than -8.

Combining the Solutions

The solution to the absolute value inequality $|x+3| < 5$ is the combination of the solutions to the two separate inequalities:

$(-\infty, 2) \cup (-8, \infty)$

This means that $x$ can take on any value less than 2 or greater than -8.

Conclusion

Solving absolute value inequalities involves rewriting the inequality as two separate inequalities and solving each one separately. By following the steps outlined in this article, you can solve absolute value inequalities of the form $|x-a| < b$. Remember to always check your solutions by plugging them back into the original inequality to ensure that they are true.

Discussion

• What are some common mistakes to avoid when solving absolute value inequalities?
• How do you handle absolute value inequalities with fractions or decimals?
• Can you think of any real-world applications of absolute value inequalities?

Additional Resources

• Khan Academy: Absolute Value Inequalities
• Mathway: Absolute Value Inequality Solver
• Wolfram Alpha: Absolute Value Inequality Calculator
  Absolute Value Inequality Q&A =============================

Frequently Asked Questions

Q: What is an absolute value inequality?

A: An absolute value inequality is a mathematical expression that involves the absolute value of a variable or expression. It is typically written in the form $|x-a| < b$, where $a$ and $b$ are constants.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to rewrite it as two separate inequalities and solve each one separately. This involves removing the absolute value sign and considering the positive and negative cases separately.

Q: What are the steps to solve an absolute value inequality?

A: The steps to solve an absolute value inequality are:

1. Rewrite the inequality as two separate inequalities.
2. Solve each inequality separately.
3. Combine the solutions to get the final answer.

Q: What are some common mistakes to avoid when solving absolute value inequalities?

A: Some common mistakes to avoid when solving absolute value inequalities include:

• Not rewriting the inequality as two separate inequalities.
• Not solving each inequality separately.
• Not combining the solutions correctly.

Q: How do I handle absolute value inequalities with fractions or decimals?

A: To handle absolute value inequalities with fractions or decimals, you need to follow the same steps as before. However, you may need to use a calculator or computer software to simplify the expressions.

Q: Can I use a calculator or computer software to solve absolute value inequalities?

A: Yes, you can use a calculator or computer software to solve absolute value inequalities. However, it's always a good idea to check your answers by plugging them back into the original inequality to ensure that they are true.

Q: What are some real-world applications of absolute value inequalities?

A: Absolute value inequalities have many real-world applications, including:

• Physics: to model the motion of objects.
• Engineering: to design and optimize systems.
• Economics: to model the behavior of economic systems.

Q: Can I use absolute value inequalities to solve systems of equations?

A: Yes, you can use absolute value inequalities to solve systems of equations. This involves using the absolute value inequality to isolate one variable and then substituting it into the other equation.

Q: How do I graph absolute value inequalities?

A: To graph absolute value inequalities, you need to plot the boundary line and then shade the region that satisfies the inequality.

Q: Can I use absolute value inequalities to solve optimization problems?

A: Yes, you can use absolute value inequalities to solve optimization problems. This involves using the absolute value inequality to find the maximum or minimum value of a function.

Additional Resources

• Khan Academy: Absolute Value Inequalities
• Mathway: Absolute Value Inequality Solver
• Wolfram Alpha: Absolute Value Inequality Calculator
• MIT OpenCourseWare: Absolute Value Inequalities
• Wolfram MathWorld: Absolute Value Inequality

Practice Problems

1. Solve the absolute value inequality $|x-2| < 3$.
2. Solve the absolute value inequality $|x+1| > 2$.
3. Solve the absolute value inequality $|x-1| < 2$.
4. Solve the absolute value inequality $|x+2| > 3$.
5. Solve the absolute value inequality $|x-3| < 4$.

Answer Key

1. $(-5, 5)$
2. $(-\infty, -3) \cup (1, \infty)$
3. $(-1, 3)$
4. $(-5, -1) \cup (5, \infty)$
5. $(-7, 7)$