Solve The Following Inequality:${ -3 \leq 5 + 2x \ \textless \ 5 }$ { [?] \leq X \ \textless \ \square \}

Mar 11, 2025 by ADMIN 113 views

Solving the Inequality: A Step-by-Step Guide

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve them. Inequalities are mathematical expressions that compare two values or expressions, and they can be used to describe a wide range of real-world situations. In this case, we will focus on solving the inequality $-3 \leq 5 + 2x < 5$ . We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Inequality

The given inequality is $-3 \leq 5 + 2x < 5$ . To solve this inequality, we need to isolate the variable $x$ . The first step is to simplify the inequality by subtracting 5 from all three parts. This gives us $-8 \leq 2x < 0$ .

Isolating the Variable

The next step is to isolate the variable $x$ . We can do this by dividing all three parts of the inequality by 2. This gives us $-4 \leq x < 0$ .

Writing the Solution in Interval Notation

The solution to the inequality can be written in interval notation as $[-4, 0)$ . This means that the value of $x$ can be any real number between $-4$ and 0, but not including 0.

Discussion

In this article, we have learned how to solve the inequality $-3 \leq 5 + 2x < 5$ . We broke down the solution into manageable steps and provided a clear explanation of each step. We also learned how to write the solution in interval notation.

Conclusion

Solving inequalities is an important skill in mathematics, and it has many real-world applications. In this article, we have provided a step-by-step guide to solving the inequality $-3 \leq 5 + 2x < 5$ . We hope that this article has been helpful in understanding how to solve inequalities and how to write the solution in interval notation.

Frequently Asked Questions

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values or expressions.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by performing the same operation on all three parts of the inequality.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality using square brackets and parentheses.

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

A: To write the solution to an inequality in interval notation, you need to use square brackets and parentheses to indicate the values that are included and excluded from the solution.

Additional Resources

Step-by-Step Solution

Step 1: Simplify the Inequality

The given inequality is $-3 \leq 5 + 2x < 5$ . To simplify the inequality, we need to subtract 5 from all three parts. This gives us $-8 \leq 2x < 0$ .

Step 2: Isolate the Variable

The next step is to isolate the variable $x$ . We can do this by dividing all three parts of the inequality by 2. This gives us $-4 \leq x < 0$ .

Step 3: Write the Solution in Interval Notation

The solution to the inequality can be written in interval notation as $[-4, 0)$ . This means that the value of $x$ can be any real number between $-4$ and 0, but not including 0.

Example Problems

Problem 1

Solve the inequality $2x + 5 \geq 11$ .

Solution

To solve the inequality, we need to isolate the variable $x$ . We can do this by subtracting 5 from all three parts. This gives us $2x \geq 6$ . Next, we can divide all three parts of the inequality by 2. This gives us $x \geq 3$ .

Problem 2

Solve the inequality $x - 2 < 7$ .

Solution

To solve the inequality, we need to isolate the variable $x$ . We can do this by adding 2 to all three parts. This gives us $x < 9$ .

Practice Problems

Problem 1

Solve the inequality $3x - 2 \geq 5$ .

Problem 2

Solve the inequality $x + 1 < 4$ .

Problem 3

Solve the inequality $2x + 1 \leq 9$ .

Problem 4

Solve the inequality $x - 3 \geq 2$ .

Problem 5

Solve the inequality $x + 2 < 1$ .

Answer Key

Problem 1

$x \geq 7/3$

Problem 2

$x < 3$

Problem 3

$x \leq 4$

Problem 4

$x \geq 5$

Problem 5

$x < -1$

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Introduction

In this article, we will answer some of the most frequently asked questions about inequalities. Inequalities are mathematical expressions that compare two values or expressions, and they can be used to describe a wide range of real-world situations. We will provide clear and concise answers to some of the most common questions about inequalities.

Q&A

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values or expressions. It can be written in the form of $a \leq b$ or $a \geq b$ , where $a$ and $b$ are expressions.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by performing the same operation on all three parts of the inequality. This can involve adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

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

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to identify the values that satisfy the inequality. You can do this by finding the values that make the inequality true and then plotting them on a number line.

Q: What is the concept of interval notation?

A: Interval notation is a way of writing the solution to an inequality using square brackets and parentheses. It is used to describe the set of values that satisfy the inequality.

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

A: To write the solution to an inequality in interval notation, you need to use square brackets and parentheses to indicate the values that are included and excluded from the solution.

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 symbol, such as $a < b$ or $a > b$ . A non-strict inequality, on the other hand, is an inequality that is written with a non-strict symbol, such as $a \leq b$ or $a \geq b$ .

Q: How do I solve a system of linear inequalities?

A: To solve a system of linear inequalities, you need to find the values that satisfy all of the inequalities in the system. You can do this by graphing the inequalities on a number line and finding the intersection of the graphs.

Additional Resources

Example Problems

Problem 1

Solve the inequality $2x + 5 \geq 11$ .

Solution

Problem 2

Solve the inequality $x - 2 < 7$ .

Solution

To solve the inequality, we need to isolate the variable $x$ . We can do this by adding 2 to all three parts. This gives us $x < 9$ .

Practice Problems

Problem 1

Solve the inequality $3x - 2 \geq 5$ .

Problem 2

Solve the inequality $x + 1 < 4$ .

Problem 3

Solve the inequality $2x + 1 \leq 9$ .

Problem 4

Solve the inequality $x - 3 \geq 2$ .

Problem 5

Solve the inequality $x + 2 < 1$ .

Answer Key

Problem 1

$x \geq 7/3$

Problem 2

$x < 3$

Problem 3

$x \leq 4$

Problem 4

$x \geq 5$

Problem 5

$x < -1$

Conclusion

In this article, we have answered some of the most frequently asked questions about inequalities. We have provided clear and concise answers to some of the most common questions about inequalities, including how to solve an inequality, how to graph an inequality on a number line, and how to write the solution to an inequality in interval notation. We hope that this article has been helpful in understanding inequalities and how to solve them.