Solve The Inequality And Graph The Solution On The Number Line Provided.$-5 + 7x \ \textgreater \ -12$Inequality Notation: $\square$Number Line: $\square \square \square \square \square \square$

Feb 26, 2025 by ADMIN 197 views

Solving and Graphing Inequalities: A Step-by-Step Guide

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Introduction

In mathematics, inequalities are a fundamental concept that helps us compare values and make decisions. Solving and graphing inequalities is an essential skill that can be applied to various real-world problems. In this article, we will focus on solving and graphing the inequality $-5 + 7x > -12$ .

Understanding the Inequality

The given inequality is $-5 + 7x > -12$ . To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The inequality sign $>$ means that the value of $x$ must be greater than the value on the other side.

Step 1: Add 5 to Both Sides

To isolate the variable $x$ , we need to get rid of the constant term $-5$ on the left side of the inequality. We can do this by adding 5 to both sides of the inequality.

-5 + 7x > -12
-5 + 5 + 7x > -12 + 5
7x > -7

Step 2: Divide Both Sides by 7

Now that we have isolated the variable $x$ , we need to get rid of the coefficient 7 on the left side of the inequality. We can do this by dividing both sides of the inequality by 7.

7x > -7
x > -7/7
x > -1

Graphing the Solution on the Number Line

Now that we have solved the inequality, we need to graph the solution on the number line. The number line is a visual representation of the solution set.

\square \square \square \square \square \square

The number line has six boxes, and we need to shade the boxes that represent the solution set. Since the inequality is $x > -1$ , we need to shade the boxes to the right of $-1$ .

\square \square \square \square \square \square

The first box represents the value $-2$ , the second box represents the value $-1$ , and so on. Since the inequality is $x > -1$ , we need to shade the boxes starting from the third box.

\square \square \square \square \square \square

The shaded boxes represent the solution set, which is $x > -1$ .

Conclusion

Solving and graphing inequalities is an essential skill that can be applied to various real-world problems. In this article, we solved the inequality $-5 + 7x > -12$ and graphed the solution on the number line. We used the steps of adding 5 to both sides and dividing both sides by 7 to isolate the variable $x$ . We then graphed the solution on the number line by shading the boxes to the right of $-1$ . The solution set is $x > -1$ , which is represented by the shaded boxes on the number line.

Tips and Tricks

When solving inequalities, always isolate the variable on one side of the inequality sign.
When graphing inequalities, always shade the boxes that represent the solution set.
When working with inequalities, always check your work by plugging in values to make sure that the solution set is correct.

Common Mistakes

Not isolating the variable on one side of the inequality sign.
Not shading the boxes that represent the solution set.
Not checking your work by plugging in values.

Real-World Applications

Solving and graphing inequalities has many real-world applications. For example, in business, inequalities can be used to compare the costs of different products or services. In engineering, inequalities can be used to compare the strengths of different materials. In economics, inequalities can be used to compare the prices of different goods and services.

Final Thoughts

Solving and graphing inequalities is an essential skill that can be applied to various real-world problems. By following the steps outlined in this article, you can solve and graph inequalities with ease. Remember to always isolate the variable on one side of the inequality sign, shade the boxes that represent the solution set, and check your work by plugging in values. With practice and patience, you can become proficient in solving and graphing inequalities.

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Introduction

In our previous article, we discussed how to solve and graph the inequality $-5 + 7x > -12$ . In this article, we will answer some frequently asked questions about solving and graphing inequalities.

Q&A

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

A: A linear inequality is an inequality that can be written in the form $ax + b > c$ , where $a$ , $b$ , and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c > 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a linear inequality?

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

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to draw a number line and shade the boxes that represent the solution set. If the inequality is of the form $x > a$ , you need to shade the boxes to the right of $a$ . If the inequality is of the form $x < a$ , you need to shade the boxes to the left of $a$ .

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict inequality sign, such as $x > a$ or $x < a$ . A non-strict inequality is an inequality that is written with a non-strict inequality sign, such as $x \geq a$ or $x \leq a$ .

Q: How do I solve a non-strict inequality?

A: To solve a non-strict inequality, you need to isolate the variable on one side of the inequality sign, just like you would with a strict inequality. However, you also need to include the value of the variable that makes the inequality true.

Q: Can I use a calculator to solve inequalities?

A: Yes, you can use a calculator to solve inequalities. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct operations.

Q: How do I check my work when solving inequalities?

A: To check your work when solving inequalities, you need to plug in values of the variable that make the inequality true and see if the inequality holds. You can also use a number line to check your work.

Tips and Tricks

Always read the inequality carefully and make sure you understand what it is asking.
Always isolate the variable on one side of the inequality sign.
Always check your work by plugging in values or using a number line.
Always use a calculator with caution and make sure it is set to the correct mode.

Common Mistakes

Not reading the inequality carefully and making sure you understand what it is asking.
Not isolating the variable on one side of the inequality sign.
Not checking your work by plugging in values or using a number line.
Not using a calculator with caution and making sure it is set to the correct mode.

Real-World Applications

Final Thoughts

Solving and graphing inequalities is an essential skill that can be applied to various real-world problems. By following the steps outlined in this article and using the tips and tricks provided, you can become proficient in solving and graphing inequalities. Remember to always read the inequality carefully, isolate the variable on one side of the inequality sign, and check your work by plugging in values or using a number line. With practice and patience, you can become proficient in solving and graphing inequalities.