Solve And Graph The Inequality: − 5 \textless 1 2 ( 2 M + 8 ) ≤ 11 -5\ \textless \ \frac{1}{2}(2m+8) \leq 11 − 5 \textless 2 1 ​ ( 2 M + 8 ) ≤ 11 Select The Correct Choice Below And Fill In Your Answer.

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Introduction

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Inequalities are mathematical expressions that compare two values, often using greater than, less than, or equal to symbols. Solving and graphing inequalities can be a challenging task, but with the right approach, it can be made easier. In this article, we will focus on solving and graphing the inequality $-5\ \textless \ \frac{1}{2}(2m+8) \leq 11$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Inequality

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The given inequality is $-5\ \textless \ \frac{1}{2}(2m+8) \leq 11$. This is a compound inequality, which means it has two parts: a less than part and a less than or equal to part. To solve this inequality, we need to isolate the variable $m$.

Step 1: Simplify the Expression

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The first step is to simplify the expression inside the parentheses. We can start by distributing the $\frac{1}{2}$ to the terms inside the parentheses:

$$\frac{1}{2}(2m+8) = m + 4$$

So, the inequality becomes:

$$-5\ \textless \ m + 4 \leq 11$$

Step 2: Subtract 4 from All Parts

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To isolate the variable $m$, we need to subtract 4 from all parts of the inequality. This will give us:

$$-5 - 4\ \textless \ m \leq 11 - 4$$

Simplifying the left-hand side, we get:

$$-9\ \textless \ m \leq 7$$

Step 3: Write the Solution in Interval Notation

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The solution to the inequality can be written in interval notation as:

$$(-9, 7]$$

This means that the value of $m$ can be any real number between -9 and 7, including 7.

Graphing the Inequality

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To graph the inequality, we need to plot the points -9 and 7 on a number line. We will use a closed circle to represent the point 7, since it is included in the solution.

Step 2: Shade the Region

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We will shade the region to the right of -9 and to the left of 7. This represents the solution to the inequality.

Conclusion

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Solving and graphing inequalities can be a challenging task, but with the right approach, it can be made easier. In this article, we solved and graphed the inequality $-5\ \textless \ \frac{1}{2}(2m+8) \leq 11$. We broke down the solution into manageable steps and provided a clear explanation of each step. We also graphed the inequality using a number line and shaded the region to represent the solution.

Frequently Asked Questions

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Q: What is the solution to the inequality $-5\ \textless \ \frac{1}{2}(2m+8) \leq 11$?

A: The solution to the inequality is $(-9, 7]$.

Q: How do I graph the inequality?

A: To graph the inequality, plot the points -9 and 7 on a number line and shade the region to the right of -9 and to the left of 7.

Q: What is the difference between a less than and a less than or equal to symbol?

A: A less than symbol (<) means that the value on the left is less than the value on the right. A less than or equal to symbol (≤) means that the value on the left is less than or equal to the value on the right.

Key Takeaways

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• Solving and graphing inequalities can be a challenging task, but with the right approach, it can be made easier.
• To solve an inequality, we need to isolate the variable.
• To graph an inequality, we need to plot the points on a number line and shade the region to represent the solution.
• The solution to an inequality can be written in interval notation.

Additional Resources

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

Final Thoughts

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Solving and graphing inequalities can be a challenging task, but with the right approach, it can be made easier. By breaking down the solution into manageable steps and providing a clear explanation of each step, we can make solving and graphing inequalities more accessible to everyone.

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Introduction

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In our previous article, we solved and graphed the inequality $-5\ \textless \ \frac{1}{2}(2m+8) \leq 11$. We broke down the solution into manageable steps and provided a clear explanation of each step. In this article, we will answer some frequently asked questions about solving and graphing inequalities.

Q&A

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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 $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, 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. 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 plot the points on a number line and shade the region to represent the solution. If the inequality is of the form $x \leq a$, you will shade the region to the left of $a$. If the inequality is of the form $x \geq a$, you will shade the region to the right of $a$.

Q: What is the difference between a less than and a less than or equal to symbol?

A: A less than symbol (<) means that the value on the left is less than the value on the right. A less than or equal to symbol (≤) means that the value on the left is less than or equal to the value on the right.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution. You can also use the quadratic formula to solve the inequality.

Q: How do I graph a quadratic inequality?

A: To graph a quadratic inequality, you need to plot the points on a number line and shade the region to represent the solution. If the inequality is of the form $x^2 + bx + c \leq 0$, you will shade the region between the roots of the quadratic equation. If the inequality is of the form $x^2 + bx + c \geq 0$, you will shade the region outside the roots of the quadratic equation.

Q: What is the difference between a compound inequality and a single inequality?

A: A compound inequality is an inequality that has two or more parts, such as $-5\ \textless \ \frac{1}{2}(2m+8) \leq 11$. A single inequality is an inequality that has only one part, such as $x \leq 5$.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to isolate the variable in each part of the inequality and then combine the solutions.

Q: How do I graph a compound inequality?

A: To graph a compound inequality, you need to plot the points on a number line and shade the region to represent the solution. You will need to shade the region between the roots of each part of the inequality.

Conclusion

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Solving and graphing inequalities can be a challenging task, but with the right approach, it can be made easier. By understanding the different types of inequalities and how to solve and graph them, you can become more confident in your ability to solve and graph inequalities.

Frequently Asked Questions

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Q: What is the solution to the inequality $-5\ \textless \ \frac{1}{2}(2m+8) \leq 11$?

A: The solution to the inequality is $(-9, 7]$.

Q: How do I graph the inequality $-5\ \textless \ \frac{1}{2}(2m+8) \leq 11$?

A: To graph the inequality, plot the points -9 and 7 on a number line and shade the region to the right of -9 and to the left of 7.

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

Key Takeaways

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• Solving and graphing inequalities can be a challenging task, but with the right approach, it can be made easier.
• To solve an inequality, you need to isolate the variable.
• To graph an inequality, you need to plot the points on a number line and shade the region to represent the solution.
• The solution to an inequality can be written in interval notation.

Additional Resources

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

Final Thoughts

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Solving and graphing inequalities can be a challenging task, but with the right approach, it can be made easier. By understanding the different types of inequalities and how to solve and graph them, you can become more confident in your ability to solve and graph inequalities.