Which Graph Represents The Solution To The Inequality 1 2 X + 15 ≥ 14 \frac{1}{2}x + 15 \geq 14 2 1 ​ X + 15 ≥ 14 ?A. (Graph A)B. (Graph B)

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Introduction

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In mathematics, inequalities are a fundamental concept that help us compare values and make decisions. When solving inequalities, we often need to find the solution set, which is the set of all possible values that satisfy the inequality. In this article, we will explore how to solve the inequality $\frac{1}{2}x + 15 \geq 14$ and determine which graph represents the solution.

Understanding the Inequality

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The given inequality is $\frac{1}{2}x + 15 \geq 14$. To solve this inequality, we need to isolate the variable $x$. We can start by subtracting 15 from both sides of the inequality:

$$\frac{1}{2}x + 15 - 15 \geq 14 - 15$$

This simplifies to:

$$\frac{1}{2}x \geq -1$$

Next, we can multiply both sides of the inequality by 2 to eliminate the fraction:

$$2 \cdot \frac{1}{2}x \geq 2 \cdot (-1)$$

This gives us:

$$x \geq -2$$

Graphing the Solution

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Now that we have solved the inequality, we can graph the solution on a number line. The solution set is all values of $x$ that are greater than or equal to -2. We can represent this on a number line by drawing a closed circle at -2 and shading the region to the right of -2.

Graph A

Graph A represents the solution to the inequality $x \geq -2$. The graph shows a closed circle at -2 and shades the region to the right of -2.

Graph B

Graph B represents the solution to the inequality $x < -2$. The graph shows an open circle at -2 and shades the region to the left of -2.

Conclusion

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Based on the solution to the inequality $\frac{1}{2}x + 15 \geq 14$, we can conclude that the correct graph is Graph A. Graph A accurately represents the solution set, which is all values of $x$ that are greater than or equal to -2.

Final Answer

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The final answer is Graph A.

Frequently Asked Questions

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Q: What is the solution to the inequality $\frac{1}{2}x + 15 \geq 14$?

A: The solution to the inequality is $x \geq -2$.

Q: Which graph represents the solution to the inequality $\frac{1}{2}x + 15 \geq 14$?

A: Graph A represents the solution to the inequality.

Q: What is the difference between Graph A and Graph B?

A: Graph A represents the solution to the inequality $x \geq -2$, while Graph B represents the solution to the inequality $x < -2$.

Step-by-Step Solution

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1. Subtract 15 from both sides of the inequality: $\frac{1}{2}x + 15 - 15 \geq 14 - 15$
2. Simplify the inequality: $\frac{1}{2}x \geq -1$
3. Multiply both sides of the inequality by 2: $2 \cdot \frac{1}{2}x \geq 2 \cdot (-1)$
4. Simplify the inequality: $x \geq -2$
5. Graph the solution on a number line: Draw a closed circle at -2 and shade the region to the right of -2.

Common Mistakes

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• Failing to isolate the variable $x$.
• Not simplifying the inequality correctly.
• Graphing the solution incorrectly.

Tips and Tricks

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• Make sure to isolate the variable $x$ correctly.
• Simplify the inequality carefully.
• Graph the solution accurately.

Real-World Applications

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Solving inequalities is an essential skill in mathematics and has many real-world applications. For example, in economics, inequalities can be used to model the relationship between variables such as supply and demand. In engineering, inequalities can be used to design and optimize systems. In finance, inequalities can be used to model risk and return.

Conclusion

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In conclusion, solving inequalities is a fundamental concept in mathematics that has many real-world applications. By understanding how to solve inequalities, we can make informed decisions and solve problems in a variety of fields. In this article, we solved the inequality $\frac{1}{2}x + 15 \geq 14$ and determined which graph represents the solution. We hope this article has provided valuable insights and practical tips for solving inequalities.

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Q&A: Solving Inequalities

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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 $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \geq 0$, 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 $x$ by adding or subtracting the same value from both sides of the inequality. You can also multiply or divide both sides of the inequality by a non-zero value.

Q: What is the solution to the inequality $x - 3 \geq 2$?

A: To solve the inequality $x - 3 \geq 2$, you need to add 3 to both sides of the inequality. This gives you $x \geq 5$.

Q: How do I graph the solution to a linear inequality?

A: To graph the solution to a linear inequality, you need to draw a closed circle at the point where the inequality is equal to zero, and then shade the region to the right or left of the point, depending on whether the inequality is greater than or less than zero.

Q: What is the solution to the inequality $x + 2 \geq 5$?

A: To solve the inequality $x + 2 \geq 5$, you need to subtract 2 from both sides of the inequality. This gives you $x \geq 3$.

Q: How do I determine which graph represents the solution to a linear inequality?

A: To determine which graph represents the solution to a linear inequality, you need to look at the graph and see which region is shaded. If the region to the right of the point is shaded, then the inequality is greater than or equal to zero. If the region to the left of the point is shaded, then the inequality is less than or equal to zero.

Q: What is the solution to the inequality $x - 2 \geq 1$?

A: To solve the inequality $x - 2 \geq 1$, you need to add 2 to both sides of the inequality. This gives you $x \geq 3$.

Q: How do I use a number line to graph the solution to a linear inequality?

A: To use a number line to graph the solution to a linear inequality, you need to draw a closed circle at the point where the inequality is equal to zero, and then shade the region to the right or left of the point, depending on whether the inequality is greater than or less than zero.

Q: What is the solution to the inequality $x + 1 \geq 4$?

A: To solve the inequality $x + 1 \geq 4$, you need to subtract 1 from both sides of the inequality. This gives you $x \geq 3$.

Q: How do I determine the solution to a linear inequality with a fraction?

A: To determine the solution to a linear inequality with a fraction, you need to multiply both sides of the inequality by the denominator of the fraction, and then solve the resulting inequality.

Q: What is the solution to the inequality $\frac{1}{2}x + 3 \geq 2$?

A: To solve the inequality $\frac{1}{2}x + 3 \geq 2$, you need to subtract 3 from both sides of the inequality, and then multiply both sides of the inequality by 2. This gives you $x \geq -4$.

Q: How do I graph the solution to a linear inequality with a fraction?

A: To graph the solution to a linear inequality with a fraction, you need to draw a closed circle at the point where the inequality is equal to zero, and then shade the region to the right or left of the point, depending on whether the inequality is greater than or less than zero.

Q: What is the solution to the inequality $\frac{1}{3}x + 2 \geq 3$?

A: To solve the inequality $\frac{1}{3}x + 2 \geq 3$, you need to subtract 2 from both sides of the inequality, and then multiply both sides of the inequality by 3. This gives you $x \geq 5$.

Q: How do I determine the solution to a linear inequality with a negative coefficient?

A: To determine the solution to a linear inequality with a negative coefficient, you need to multiply both sides of the inequality by -1, and then solve the resulting inequality.

Q: What is the solution to the inequality $-x + 2 \geq 1$?

A: To solve the inequality $-x + 2 \geq 1$, you need to subtract 2 from both sides of the inequality, and then multiply both sides of the inequality by -1. This gives you $x \leq 1$.

Q: How do I graph the solution to a linear inequality with a negative coefficient?

A: To graph the solution to a linear inequality with a negative coefficient, you need to draw a closed circle at the point where the inequality is equal to zero, and then shade the region to the left or right of the point, depending on whether the inequality is less than or greater than zero.

Q: What is the solution to the inequality $-x - 2 \geq 3$?

A: To solve the inequality $-x - 2 \geq 3$, you need to add 2 to both sides of the inequality, and then multiply both sides of the inequality by -1. This gives you $x \leq -5$.

Q: How do I determine the solution to a linear inequality with a variable on both sides?

A: To determine the solution to a linear inequality with a variable on both sides, you need to add or subtract the same value from both sides of the inequality, and then solve the resulting inequality.

Q: What is the solution to the inequality $x + 2 \geq x - 1$?

A: To solve the inequality $x + 2 \geq x - 1$, you need to subtract $x$ from both sides of the inequality, and then add 1 to both sides of the inequality. This gives you $3 \geq -1$.

Q: How do I graph the solution to a linear inequality with a variable on both sides?

A: To graph the solution to a linear inequality with a variable on both sides, you need to draw a closed circle at the point where the inequality is equal to zero, and then shade the region to the right or left of the point, depending on whether the inequality is greater than or less than zero.

Q: What is the solution to the inequality $x - 2 \geq x + 1$?

A: To solve the inequality $x - 2 \geq x + 1$, you need to subtract $x$ from both sides of the inequality, and then add 2 to both sides of the inequality. This gives you $-3 \geq x$.

Q: How do I determine the solution to a linear inequality with a fraction and a variable on both sides?

A: To determine the solution to a linear inequality with a fraction and a variable on both sides, you need to multiply both sides of the inequality by the denominator of the fraction, and then solve the resulting inequality.

Q: What is the solution to the inequality $\frac{1}{2}x + 2 \geq \frac{1}{2}x - 1$?

A: To solve the inequality $\frac{1}{2}x + 2 \geq \frac{1}{2}x - 1$, you need to subtract $\frac{1}{2}x$ from both sides of the inequality, and then add 1 to both sides of the inequality. This gives you $3 \geq -1$.

Q: How do I graph the solution to a linear inequality with a fraction and a variable on both sides?

A: To graph the solution to a linear inequality with a fraction and a variable on both sides, you need to draw a closed circle at the point where the inequality is equal to zero, and then shade the region to the right or left of the point, depending on whether the inequality is greater than or less than zero.

Q: What is the solution to the inequality $\frac{1}{3}x + 2 \geq \frac{1}{3}x - 1$?

A: To solve the inequality $\frac{1}{3}x + 2 \geq \frac{1}{3}x - 1$, you need to subtract $\frac{1}{3}x$ from both sides of the inequality, and then add 1 to both sides of the inequality. This gives you $3 \geq -1$.

Q: How do I determine the solution to a linear inequality with a negative coefficient and a variable on both sides?

A: To determine the solution to a linear inequality with a negative coefficient and a variable on both sides, you need to multiply both sides of the inequality by -1, and then solve the resulting inequality.

Q: What is the solution to the inequality $-x + 2 \geq -x - 1$?

A