Graphing the Solution on the Number Line

Oct 13, 2025 by miodeya.com 109 views

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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 $ax + b < c$ , where $a$ , $b$ , and $c$ are constants. We will use the given inequality $-12 < \frac{x+177}{-13}$ as an example to demonstrate the step-by-step process of solving linear inequalities.

Step 1: Multiply Both Sides by the Denominator

The first step in solving the inequality is to eliminate the fraction by multiplying both sides by the denominator, which is $-13$ . However, we must remember to reverse the direction of the inequality when multiplying by a negative number.

-12 < \frac{x+177}{-13}

Multiplying both sides by $-13$ gives us:

-12(-13) < (x+177)

Simplifying the left-hand side, we get:

156 < x+177

Step 2: Subtract 177 from Both Sides

The next step is to isolate the variable $x$ by subtracting 177 from both sides of the inequality.

156 < x+177

Subtracting 177 from both sides gives us:

-21 < x

Step 3: Write the Solution in Inequality Notation

The final step is to write the solution in inequality notation. Since $x$ is greater than $-21$ , we can write the solution as:

x > -21

Graphing the Solution on the Number Line

To visualize the solution, we can graph it on the number line. The number line consists of a series of points, each representing a value of $x$ . We can plot the point $-21$ on the number line and shade the region to the right of it, indicating that $x$ is greater than $-21$ .

Conclusion

Solving linear inequalities requires a step-by-step approach, starting with multiplying both sides by the denominator and reversing the direction of the inequality when necessary. By following these steps, we can isolate the variable and write the solution in inequality notation. Graphing the solution on the number line provides a visual representation of the solution and helps to reinforce the concept.

Tips and Tricks

When multiplying both sides by a negative number, remember to reverse the direction of the inequality.
When subtracting a constant from both sides, make sure to subtract the same value from both sides.
When writing the solution in inequality notation, use the correct symbols (e.g., $>$ , $<$ , $\geq$ , $\leq$ ) to indicate the direction of the inequality.

Common Mistakes to Avoid

Failing to reverse the direction of the inequality when multiplying by a negative number.
Subtracting the wrong value from both sides.
Writing the solution in inequality notation with the wrong symbols.

Real-World Applications

Linear inequalities have numerous real-world applications, including:

Finance: Calculating interest rates and investment returns.
Science: Modeling population growth and decay.
Engineering: Designing and optimizing systems.

Practice Problems

Try solving the following linear inequalities:

$2x + 5 < 11$
$\frac{x-3}{4} > 2$
$x - 2 > 7$

Solutions

$2x + 5 < 11$ $2 x + 5 < 11$
- Subtract 5 from both sides: $2x < 6$
- Divide both sides by 2: $x < 3$
$\frac{x-3}{4} > 2$ $\frac{x - 3}{4} > 2$
- Multiply both sides by 4: $x - 3 > 8$
- Add 3 to both sides: $x > 11$
$x - 2 > 7$ $x - 2 > 7$
- Add 2 to both sides: $x > 9$

Conclusion

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Introduction

In our previous article, we explored the step-by-step process of solving linear inequalities. However, we understand that sometimes, it's easier to learn through questions and answers. In this article, we'll address some common questions and provide detailed explanations to help you better understand solving linear inequalities.

Q&A

Q1: What is the first step in solving a linear inequality?

A1: The first step in solving a linear inequality is to eliminate the fraction by multiplying both sides by the denominator. However, if the denominator is negative, we must reverse the direction of the inequality.

Q2: How do I know when to reverse the direction of the inequality?

A2: You should reverse the direction of the inequality when multiplying both sides by a negative number. This is because multiplying a negative number by a positive number results in a negative number, and vice versa.

Q3: What if I have a fraction with a variable in the numerator and a constant in the denominator?

A3: In this case, you can multiply both sides by the constant in the denominator to eliminate the fraction. However, make sure to reverse the direction of the inequality if the constant is negative.

Q4: Can I add or subtract the same value from both sides of an inequality?

A4: Yes, you can add or subtract the same value from both sides of an inequality. However, make sure to do so consistently, either adding or subtracting the same value from both sides.

Q5: How do I write the solution in inequality notation?

A5: To write the solution in inequality notation, use the correct symbols (e.g., $>$ , $<$ , $\geq$ , $\leq$ ) to indicate the direction of the inequality. For example, if the solution is $x > 5$ , you would write it as $x > 5$ .

Common Mistakes to Avoid

Failing to reverse the direction of the inequality when multiplying by a negative number.
Subtracting the wrong value from both sides.
Writing the solution in inequality notation with the wrong symbols.

Real-World Applications

Linear inequalities have numerous real-world applications, including:

Finance: Calculating interest rates and investment returns.
Science: Modeling population growth and decay.
Engineering: Designing and optimizing systems.

Practice Problems

Try solving the following linear inequalities:

$\frac{x+2}{3} < 4$
$2x - 5 > 3$
$\frac{x-1}{2} \geq 3$

Solutions

$\frac{x+2}{3} < 4$ $\frac{x + 2}{3} < 4$
- Multiply both sides by 3: $x + 2 < 12$
- Subtract 2 from both sides: $x < 10$
$2x - 5 > 3$ $2 x - 5 > 3$
- Add 5 to both sides: $2x > 8$
- Divide both sides by 2: $x > 4$
$\frac{x-1}{2} \geq 3$ $\frac{x - 1}{2} \geq 3$
- Multiply both sides by 2: $x - 1 \geq 6$
- Add 1 to both sides: $x \geq 7$

Conclusion

Additional Resources

For more practice problems and examples, check out the following resources:

Khan Academy: Linear Inequalities
Mathway: Linear Inequality Solver
IXL: Linear Inequalities

Final Tips

Practice, practice, practice! The more you practice solving linear inequalities, the more comfortable you'll become with the process.
Use online resources, such as Khan Academy and Mathway, to supplement your learning.
Don't be afraid to ask for help if you're struggling with a particular concept or problem.