Solve Each Inequality For The Unknown Value.a. 13 + 4 X \textgreater 9 13 + 4x \ \textgreater \ 9 13 + 4 X \textgreater 9 Enter The Answer In The Space Provided. Use Numbers Instead Of Words.

Mar 10, 2025 by ADMIN 194 views

Solving Linear Inequalities

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Introduction

Linear inequalities are mathematical expressions that contain an inequality sign and one or more variables. In this article, we will focus on solving linear inequalities for the unknown value. We will use the given inequality $13 + 4x \ \textgreater \ 9$ as an example.

Understanding the Inequality

The given inequality is $13 + 4x \ \textgreater \ 9$ . To solve this inequality, we need to isolate the variable $x$ . The first step is to subtract 13 from both sides of the inequality.

Subtracting 13 from Both Sides

Subtracting 13 from both sides of the inequality gives us:

4x \ \textgreater \ -4

Dividing Both Sides by 4

Next, we need to divide both sides of the inequality by 4 to isolate the variable $x$ .

x \ \textgreater \ -1

Conclusion

In conclusion, the solution to the inequality $13 + 4x \ \textgreater \ 9$ is $x \ \textgreater \ -1$ . This means that any value of $x$ that is greater than -1 will satisfy the inequality.

Solving Another Inequality

Let's solve another inequality to reinforce our understanding of solving linear inequalities.

Inequality 2

The given inequality is $7 - 3x \ \textless \ 2$ . To solve this inequality, we need to isolate the variable $x$ .

Subtracting 7 from Both Sides

Subtracting 7 from both sides of the inequality gives us:

-3x \ \textless \ -5

Dividing Both Sides by -3

Next, we need to divide both sides of the inequality by -3 to isolate the variable $x$ . When we divide by a negative number, we need to reverse the direction of the inequality sign.

x \ \textgreater \ \frac{5}{3}

Conclusion

In conclusion, the solution to the inequality $7 - 3x \ \textless \ 2$ is $x \ \textgreater \ \frac{5}{3}$ . This means that any value of $x$ that is greater than $\frac{5}{3}$ will satisfy the inequality.

Solving a System of Linear Inequalities

A system of linear inequalities is a set of two or more linear inequalities that are combined using the logical operators "and" and "or". Let's solve a system of two linear inequalities.

Inequality 1

The given inequality is $x + 2y \ \textless \ 4$ . To solve this inequality, we need to isolate the variable $x$ .

Subtracting 2y from Both Sides

Subtracting 2y from both sides of the inequality gives us:

x \ \textless \ 4 - 2y

Inequality 2

The given inequality is $2x - 3y \ \textgreater \ 1$ . To solve this inequality, we need to isolate the variable $x$ .

Adding 3y to Both Sides

Adding 3y to both sides of the inequality gives us:

2x \ \textgreater \ 1 + 3y

Dividing Both Sides by 2

Next, we need to divide both sides of the inequality by 2 to isolate the variable $x$ .

x \ \textgreater \ \frac{1 + 3y}{2}

Conclusion

In conclusion, the solution to the system of linear inequalities $x + 2y \ \textless \ 4$ and $2x - 3y \ \textgreater \ 1$ is the set of all points $(x, y)$ that satisfy both inequalities.

Solving a Linear Inequality with Absolute Value

A linear inequality with absolute value is an inequality that contains an absolute value expression. Let's solve a linear inequality with absolute value.

Inequality 3

The given inequality is $|x| \ \textless \ 2$ . To solve this inequality, we need to isolate the variable $x$ .

Removing the Absolute Value

Removing the absolute value from the inequality gives us two separate inequalities:

x \ \textless \ 2 \ \text{and} \ x \ \textgreater \ -2

Conclusion

In conclusion, the solution to the linear inequality with absolute value $|x| \ \textless \ 2$ is the set of all points $x$ that satisfy both inequalities $x \ \textless \ 2$ and $x \ \textgreater \ -2$ .

Solving a Linear Inequality with Fractions

A linear inequality with fractions is an inequality that contains a fraction. Let's solve a linear inequality with fractions.

Inequality 4

The given inequality is $\frac{x}{2} \ \textless \ 3$ . To solve this inequality, we need to isolate the variable $x$ .

Multiplying Both Sides by 2

Multiplying both sides of the inequality by 2 gives us:

x \ \textless \ 6

Conclusion

In conclusion, the solution to the linear inequality with fractions $\frac{x}{2} \ \textless \ 3$ is the set of all points $x$ that satisfy the inequality $x \ \textless \ 6$ .

Solving a Linear Inequality with Decimals

A linear inequality with decimals is an inequality that contains a decimal. Let's solve a linear inequality with decimals.

Inequality 5

The given inequality is $2.5x \ \textless \ 5$ . To solve this inequality, we need to isolate the variable $x$ .

Dividing Both Sides by 2.5

Dividing both sides of the inequality by 2.5 gives us:

x \ \textless \ 2

Conclusion

In conclusion, the solution to the linear inequality with decimals $2.5x \ \textless \ 5$ is the set of all points $x$ that satisfy the inequality $x \ \textless \ 2$ .

Solving a Linear Inequality with Negative Numbers

A linear inequality with negative numbers is an inequality that contains a negative number. Let's solve a linear inequality with negative numbers.

Inequality 6

The given inequality is $-3x \ \textless \ -6$ . To solve this inequality, we need to isolate the variable $x$ .

Dividing Both Sides by -3

Dividing both sides of the inequality by -3 gives us:

x \ \textgreater \ 2

Conclusion

In conclusion, the solution to the linear inequality with negative numbers $-3x \ \textless \ -6$ is the set of all points $x$ that satisfy the inequality $x \ \textgreater \ 2$ .

Solving a Linear Inequality with Exponents

A linear inequality with exponents is an inequality that contains an exponent. Let's solve a linear inequality with exponents.

Inequality 7

The given inequality is $x^2 \ \textless \ 4$ . To solve this inequality, we need to isolate the variable $x$ .

Taking the Square Root of Both Sides

Taking the square root of both sides of the inequality gives us:

x \ \textless \ 2 \ \text{and} \ x \ \textgreater \ -2

Conclusion

In conclusion, the solution to the linear inequality with exponents $x^2 \ \textless \ 4$ is the set of all points $x$ that satisfy both inequalities $x \ \textless \ 2$ and $x \ \textgreater \ -2$ .

Solving a Linear Inequality with Absolute Value and Fractions

A linear inequality with absolute value and fractions is an inequality that contains an absolute value expression and a fraction. Let's solve a linear inequality with absolute value and fractions.

Inequality 8

The given inequality is $|\frac{x}{2}| \ \textless \ 2$ . To solve this inequality, we need to isolate the variable $x$ .

Removing the Absolute Value

Removing the absolute value from the inequality gives us two separate inequalities:

\frac{x}{2} \ \textless \ 2 \ \text{and} \ \frac{x}{2} \ \textgreater \ -2

Multiplying Both Sides by 2

Multiplying both sides of the inequality by 2 gives us:

x \ \textless \ 4 \ \text{and} \ x \ \textgreater \ -4

Conclusion

In conclusion, the solution to the linear inequality with absolute value and fractions $|\frac{x}{2}| \ \textless \ 2$ is the set of all points $

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Introduction

In the previous article, we discussed how to solve linear inequalities. In this article, we will answer some frequently asked questions about solving linear inequalities.

Q: What is a linear inequality?

A linear inequality is a mathematical expression that contains an inequality sign and one or more variables. It is a statement that two expressions are not equal, but one is greater than or less than the other.

Q: How do I solve a linear inequality?

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

A linear equation is a statement that two expressions are equal, while a linear inequality is a statement that two expressions are not equal, but one is greater than or less than the other.

Q: How do I solve a linear inequality with absolute value?

To solve a linear inequality with absolute value, you need to remove the absolute value sign and solve the resulting inequality. You may need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative.

Q: How do I solve a linear inequality with fractions?

To solve a linear inequality with fractions, you need to multiply or divide both sides of the inequality by the denominator of the fraction. This will eliminate the fraction and allow you to solve the inequality.

Q: How do I solve a linear inequality with decimals?

To solve a linear inequality with decimals, you need to multiply or divide both sides of the inequality by the decimal value. This will eliminate the decimal and allow you to solve the inequality.

Q: How do I solve a linear inequality with negative numbers?

To solve a linear inequality with negative numbers, you need to multiply or divide both sides of the inequality by the negative number. This will change the direction of the inequality sign.

Q: How do I solve a linear inequality with exponents?

To solve a linear inequality with exponents, you need to take the square root of both sides of the inequality. This will eliminate the exponent and allow you to solve the inequality.

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

Some common mistakes to avoid when solving linear inequalities include:

Not isolating the variable on one side of the inequality sign
Not considering two cases when solving a linear inequality with absolute value
Not multiplying or dividing both sides of the inequality by the same non-zero value
Not changing the direction of the inequality sign when multiplying or dividing by a negative number

Q: How do I check my solution to a linear inequality?

To check your solution to a linear inequality, you need to plug in the value of the variable into the original inequality and see if it is true. If it is true, then your solution is correct. If it is not true, then your solution is incorrect.

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

Linear inequalities have many real-world applications, including:

Budgeting and finance
Science and engineering
Economics and business
Computer science and programming

Q: How do I use linear inequalities in real-world problems?

To use linear inequalities in real-world problems, you need to identify the variables and the inequality sign, and then solve the inequality to find the solution. You can use linear inequalities to make decisions, optimize resources, and solve problems.

Conclusion

In conclusion, solving linear inequalities is an important skill that has many real-world applications. By understanding how to solve linear inequalities, you can make informed decisions, optimize resources, and solve problems. Remember to avoid common mistakes, check your solution, and use linear inequalities in real-world problems.