Subtraction Property of Inequality

Oct 13, 2025 by miodeya.com 91 views

Introduction

Inequalities are mathematical expressions that compare two values, indicating whether one is greater than, less than, or equal to the other. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will walk through the steps of solving the inequality $7x + 4 \ \textless \ 46$ and identify the property used in each step.

Step 1: Given Inequality

The given inequality is $7x + 4 \ \textless \ 46$ . This is the starting point of our solution.

Property Used: None

At this stage, we have not applied any properties to the inequality. We are simply stating the given inequality.

Step 2: Subtracting 4 from Both Sides

To isolate the term with the variable, we need to subtract 4 from both sides of the inequality.

$7x + 4 \ \textless \ 46$

Subtracting 4 from both sides gives us:

$7x \ \textless \ 42$

Property Used: Subtraction Property of Inequality

The subtraction property of inequality states that if $a \ \textless \ b$ , then $a - c \ \textless \ b - c$ for any real number $c$ . In this case, we subtracted 4 from both sides of the inequality.

Step 3: Dividing Both Sides by 7

To solve for $x$ , we need to divide both sides of the inequality by 7.

$7x \ \textless \ 42$

Dividing both sides by 7 gives us:

$x \ \textless \ 6$

Property Used: Division Property of Inequality

The division property of inequality states that if $a \ \textless \ b$ and $b \neq 0$ , then $\frac{a}{b} \ \textless \ \frac{b}{b}$ for any real number $b$ . In this case, we divided both sides of the inequality by 7.

Conclusion

In this article, we solved the inequality $7x + 4 \ \textless \ 46$ by applying the subtraction and division properties of inequality. We identified the property used in each step and provided a clear explanation of the solution process.

Key Takeaways

The subtraction property of inequality states that if $a \ \textless \ b$ , then $a - c \ \textless \ b - c$ for any real number $c$ .
The division property of inequality states that if $a \ \textless \ b$ and $b \neq 0$ , then $\frac{a}{b} \ \textless \ \frac{b}{b}$ for any real number $b$ .
Solving inequalities involves isolating the variable on one side of the inequality sign.

Practice Problems

Solve the inequality $3x - 2 \ \textless \ 11$ .
Solve the inequality $x + 5 \ \textless \ 17$ .

Answer Key

$x \ \textless \ 5.33$
$x \ \textless \ 12$

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Intermediate Algebra" by Charles P. McKeague

Introduction

In our previous article, we walked through the steps of solving the inequality $7x + 4 \ \textless \ 46$ and identified the property used in each step. In this article, we will answer some frequently asked questions about solving inequalities.

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to read and understand the given inequality. This involves identifying the variable, the inequality sign, and the constant term.

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 $ax + b \ \textless \ c$ , where $a$ , $b$ , and $c$ are constants and $x$ is the variable. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c \ \textless \ 0$ , where $a$ , $b$ , and $c$ are constants and $x$ is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can follow these steps:

Read and understand the given inequality.
Isolate the variable on one side of the inequality sign.
Use the properties of inequality to simplify the inequality.
Write the solution in interval notation.

Q: What are the properties of inequality?

A: The properties of inequality are:

Addition Property of Inequality: If $a \ \textless \ b$ , then $a + c \ \textless \ b + c$ for any real number $c$ .
Subtraction Property of Inequality: If $a \ \textless \ b$ , then $a - c \ \textless \ b - c$ for any real number $c$ .
Multiplication Property of Inequality: If $a \ \textless \ b$ and $c > 0$ , then $ac \ \textless \ bc$ .
Division Property of Inequality: If $a \ \textless \ b$ and $b \neq 0$ , then $\frac{a}{b} \ \textless \ \frac{b}{b}$ for any real number $b$ .

Q: How do I write the solution in interval notation?

A: To write the solution in interval notation, you need to identify the values of $x$ that satisfy the inequality. If the inequality is of the form $x \ \textless \ a$ , then the solution is $(-\infty, a)$ . If the inequality is of the form $x \ \textgreater \ a$ , then the solution is $(a, \infty)$ .

Q: What is the difference between a solution set and a graph?

A: A solution set is a set of values that satisfy the inequality, while a graph is a visual representation of the inequality.

Q: How do I graph an inequality?

A: To graph an inequality, you can follow these steps:

Identify the values of $x$ that satisfy the inequality.
Plot the values on a number line.
Shade the region that satisfies the inequality.

Conclusion

In this article, we answered some frequently asked questions about solving inequalities. We covered topics such as the first step in solving an inequality, the difference between a linear inequality and a quadratic inequality, and how to graph an inequality.

Key Takeaways

The first step in solving an inequality is to read and understand the given inequality.
The properties of inequality are addition, subtraction, multiplication, and division.
To write the solution in interval notation, you need to identify the values of $x$ that satisfy the inequality.
A solution set is a set of values that satisfy the inequality, while a graph is a visual representation of the inequality.

Practice Problems

Solve the inequality $2x + 3 \ \textless \ 5$ .
Solve the inequality $x - 2 \ \textless \ 3$ .

Answer Key

$x \ \textless \ 1$
$x \ \textless \ 5$

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Intermediate Algebra" by Charles P. McKeague

Note: The references provided are for general information purposes only and are not directly related to the specific inequality solved in this article.