Solve The Inequality: 3 + X \textless 5 3 + X \ \textless \ 5 3 + X \textless 5

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Introduction

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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 the inequality $3 + x < 5$, which is a simple yet essential example of a linear inequality. We will break down the solution step by step, using clear and concise language to ensure that readers understand the concept.

What is a Linear Inequality?

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A linear inequality is an inequality that can be written in the form $ax + b < c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Linear inequalities can be solved using various methods, including algebraic manipulation and graphical representation.

Solving the Inequality $3 + x < 5$

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To solve the inequality $3 + x < 5$, we need to isolate the variable $x$. We can do this by subtracting $3$ from both sides of the inequality.

Step 1: Subtract 3 from Both Sides

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Subtracting $3$ from both sides of the inequality gives us:

$$x < 5 - 3$$

Step 2: Simplify the Right-Hand Side

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Simplifying the right-hand side of the inequality gives us:

$$x < 2$$

Step 3: Write the Solution in Interval Notation

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The solution to the inequality $x < 2$ can be written in interval notation as:

$$(-\infty, 2)$$

This means that the solution is all real numbers less than $2$.

Graphical Representation

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The inequality $x < 2$ can also be represented graphically on a number line. The number line is divided into two parts: the part to the left of $2$ and the part to the right of $2$. The solution to the inequality is the part to the left of $2$.

Conclusion

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Solving linear inequalities is an essential skill for students to master. In this article, we have solved the inequality $3 + x < 5$ using algebraic manipulation and graphical representation. We have shown that the solution to the inequality is $x < 2$, which can be written in interval notation as $(-\infty, 2)$. We hope that this article has provided a clear and concise explanation of how to solve linear inequalities.

Frequently Asked Questions

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Q: What is a linear 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, and $x$ is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable $x$ by adding or subtracting the same value from both sides of the inequality.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality using a specific notation. For example, the solution to the inequality $x < 2$ can be written in interval notation as $(-\infty, 2)$.

Further Reading

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If you want to learn more about solving linear inequalities, we recommend checking out the following resources:

• Khan Academy: Linear Inequalities
• Mathway: Solving Linear Inequalities
• Wolfram Alpha: Linear Inequalities

References

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• [1] "Linear Inequalities" by Khan Academy
• [2] "Solving Linear Inequalities" by Mathway
• [3] "Linear Inequalities" by Wolfram Alpha
  

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Introduction

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In our previous article, we discussed how to solve linear inequalities using algebraic manipulation and graphical representation. However, we know that solving linear inequalities can be a challenging task, especially for students who are new to the concept. In this article, we will provide a Q&A guide to help students understand the concept of linear inequalities and how to solve them.

Q&A Guide

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Q: What is a linear 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, and $x$ is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable $x$ by adding or subtracting the same value from both sides of the inequality.

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

A: A linear equation is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form $ax + b < c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

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

A: To graph a linear inequality on a number line, you need to identify the critical point, which is the value of $x$ that makes the inequality true. Then, you need to shade the region to the left or right of the critical point, depending on whether the inequality is less than or greater than.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality using a specific notation. For example, the solution to the inequality $x < 2$ can be written in interval notation as $(-\infty, 2)$.

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 identify the critical point and the direction of the inequality. Then, you need to use the following notation:

• $(-\infty, a)$: all real numbers less than $a$
• $(a, \infty)$: all real numbers greater than $a$
• $(-\infty, a]$ or $[a, \infty)$: all real numbers less than or equal to $a$ or greater than or equal to $a$

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

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

A: To solve a system of linear inequalities, you need to find the solution to each inequality separately and then find the intersection of the solutions.

Examples

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Example 1: Solving a Linear Inequality

Solve the inequality $2x + 3 < 5$.

Step 1: Subtract 3 from both sides

Subtracting 3 from both sides of the inequality gives us:

$$2x < 5 - 3$$

Step 2: Simplify the right-hand side

Simplifying the right-hand side of the inequality gives us:

$$2x < 2$$

Step 3: Divide both sides by 2

Dividing both sides of the inequality by 2 gives us:

$$x < 1$$

Example 2: Graphing a Linear Inequality

Graph the inequality $x - 2 < 3$ on a number line.

Step 1: Identify the critical point

The critical point is the value of $x$ that makes the inequality true. In this case, the critical point is $x = 5$.

Step 2: Shade the region

Shading the region to the left of the critical point gives us:

Example 3: Writing the Solution in Interval Notation

Write the solution to the inequality $x < 2$ in interval notation.

Step 1: Identify the critical point

The critical point is the value of $x$ that makes the inequality true. In this case, the critical point is $x = 2$.

Step 2: Use interval notation

Using interval notation, we can write the solution as $(-\infty, 2)$.

Conclusion

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Solving linear inequalities can be a challenging task, but with practice and patience, you can master the concept. In this article, we have provided a Q&A guide to help students understand the concept of linear inequalities and how to solve them. We have also provided examples to illustrate the concept. We hope that this article has been helpful in providing a clear and concise explanation of how to solve linear inequalities.

Frequently Asked Questions

---

Q: What is a linear 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, and $x$ is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable $x$ by adding or subtracting the same value from both sides of the inequality.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality using a specific notation. For example, the solution to the inequality $x < 2$ can be written in interval notation as $(-\infty, 2)$.

Further Reading

---

If you want to learn more about solving linear inequalities, we recommend checking out the following resources:

• Khan Academy: Linear Inequalities
• Mathway: Solving Linear Inequalities
• Wolfram Alpha: Linear Inequalities

References

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• [1] "Linear Inequalities" by Khan Academy
• [2] "Solving Linear Inequalities" by Mathway
• [3] "Linear Inequalities" by Wolfram Alpha