Solve The Linear Inequality. Express The Solution Using Interval Notation. 2 ( 9 X − 1 ) ≤ 16 X + 34 2(9x - 1) \leq 16x + 34 2 ( 9 X − 1 ) ≤ 16 X + 34 Graph The Solution Set.

Mar 8, 2025 by ADMIN 175 views

Solving Linear Inequalities: A Step-by-Step Guide

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Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving linear inequalities of the form $ax \leq b$ , where $a$ and $b$ are constants, and $x$ is the variable. We will also explore how to express the solution using interval notation and graph the solution set.

What is a Linear Inequality?

A linear inequality is an inequality that can be written in the form $ax \leq b$ , where $a$ and $b$ are constants, and $x$ is the variable. The inequality can be either less than or equal to ( $\leq$ ) or greater than or equal to ( $\geq$ ). For example, $2x - 3 \leq 5$ is a linear inequality.

Solving Linear Inequalities

To solve a linear inequality, we need to isolate the variable $x$ on one side of the inequality. We can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides by the same non-zero value.

Step 1: Simplify the Inequality

The first step in solving a linear inequality is to simplify the inequality by combining like terms. For example, if we have the inequality $2(9x - 1) \leq 16x + 34$ , we can simplify it by distributing the 2 to the terms inside the parentheses: $18x - 2 \leq 16x + 34$ .

Step 2: Add or Subtract the Same Value to Both Sides

Next, we need to add or subtract the same value to both sides of the inequality to isolate the variable $x$ . In this case, we can add 2 to both sides of the inequality to get: $18x \leq 16x + 36$ .

Step 3: Subtract the Same Value from Both Sides

Now, we need to subtract the same value from both sides of the inequality to isolate the variable $x$ . In this case, we can subtract $16x$ from both sides of the inequality to get: $2x \leq 36$ .

Step 4: Divide Both Sides by the Same Non-Zero Value

Finally, we need to divide both sides of the inequality by the same non-zero value to solve for $x$ . In this case, we can divide both sides of the inequality by 2 to get: $x \leq 18$ .

Expressing the Solution Using Interval Notation

Once we have solved the linear inequality, we need to express the solution using interval notation. Interval notation is a way of writing the solution set in a compact and concise manner.

For example, if we have the inequality $x \leq 18$ , we can express the solution set in interval notation as $(-\infty, 18]$ .

Graphing the Solution Set

Graphing the solution set is an important step in solving linear inequalities. The solution set is the set of all values of $x$ that satisfy the inequality.

For example, if we have the inequality $x \leq 18$ , we can graph the solution set by drawing a line at $x = 18$ and shading the region to the left of the line.

Conclusion

Example Problems

Problem 1

Solve the linear inequality $3x - 2 \leq 5x + 1$ and express the solution set using interval notation.

Solution

To solve the linear inequality, we need to isolate the variable $x$ on one side of the inequality. We can do this by subtracting $3x$ from both sides of the inequality to get: $-2 \leq 2x + 1$ . Next, we need to subtract 1 from both sides of the inequality to get: $-3 \leq 2x$ . Finally, we need to divide both sides of the inequality by 2 to get: $-1.5 \leq x$ .

The solution set can be expressed in interval notation as $[-1.5, \infty)$ .

Problem 2

Solve the linear inequality $2x + 3 \geq 5x - 2$ and graph the solution set.

Solution

To solve the linear inequality, we need to isolate the variable $x$ on one side of the inequality. We can do this by subtracting $2x$ from both sides of the inequality to get: $3 \geq 3x - 2$ . Next, we need to add 2 to both sides of the inequality to get: $5 \geq 3x$ . Finally, we need to divide both sides of the inequality by 3 to get: $\frac{5}{3} \geq x$ .

The solution set can be graphed by drawing a line at $x = \frac{5}{3}$ and shading the region to the right of the line.

Final Thoughts

Solving linear inequalities is a crucial skill for students to master. By following the steps outlined in this article, we can solve linear inequalities and express the solution set using interval notation. We can also graph the solution set to visualize the solution set. With practice and patience, students can become proficient in solving linear inequalities and apply this skill to a wide range of mathematical problems.

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Q&A: Solving Linear Inequalities

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form $ax \leq b$ , where $a$ and $b$ are constants, and $x$ is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable $x$ on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides by the same non-zero value.

Q: What is interval notation?

A: Interval notation is a way of writing the solution set in a compact and concise manner. It is used to express the solution set of a linear inequality.

Q: How do I graph the solution set of a linear inequality?

A: To graph the solution set of a linear inequality, you need to draw a line at the boundary of the solution set and shade the region that satisfies the inequality.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear equation is an equation that can be written in the form $ax = b$ , where $a$ and $b$ are constants, and $x$ is the variable. A linear inequality is an inequality that can be written in the form $ax \leq b$ , where $a$ and $b$ are constants, and $x$ is the variable.

Q: Can I use the same steps to solve a linear inequality as I would to solve a linear equation?

A: No, you cannot use the same steps to solve a linear inequality as you would to solve a linear equation. Linear inequalities require a different set of steps to solve.

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

A: Some common mistakes to avoid when solving linear inequalities include:

Not isolating the variable $x$ on one side of the inequality
Not checking the direction of the inequality
Not considering the boundary of the solution set

Q: How do I check my work when solving a linear inequality?

A: To check your work when solving a linear inequality, you need to plug in a value of $x$ that satisfies the inequality and make sure that the inequality holds true.

Q: Can I use a calculator to solve linear inequalities?

A: Yes, you can use a calculator to solve linear inequalities. However, you need to make sure that you are using the correct steps and that you are checking your work.

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

A: Linear inequalities have many real-world applications, including:

Finance: Linear inequalities are used to calculate interest rates and investment returns.
Science: Linear inequalities are used to model population growth and decay.
Engineering: Linear inequalities are used to design and optimize systems.

Conclusion

Solving linear inequalities is a crucial skill for students to master. By following the steps outlined in this article, you can solve linear inequalities and express the solution set using interval notation. You can also graph the solution set to visualize the solution set. With practice and patience, you can become proficient in solving linear inequalities and apply this skill to a wide range of mathematical problems.

Additional Resources

Khan Academy: Linear Inequalities
Mathway: Linear Inequalities
Wolfram Alpha: Linear Inequalities

Final Thoughts

Solving linear inequalities is a challenging but rewarding topic. By mastering the skills outlined in this article, you can become proficient in solving linear inequalities and apply this skill to a wide range of mathematical problems. Remember to practice regularly and to check your work to ensure that you are getting the correct answers.