What Is The Solution To − 3 4 X + 2 ≤ − 7 -\frac{3}{4} X + 2 \leq -7 − 4 3 ​ X + 2 ≤ − 7 ?A. X ≥ − 12 X \geq -12 X ≥ − 12 B. X ≥ 12 X \geq 12 X ≥ 12 C. X ≤ 12 X \leq 12 X ≤ 12 D. X ≤ − 12 X \leq -12 X ≤ − 12

Introduction

In mathematics, solving inequalities is a crucial aspect of algebra and is used to find the range of values that a variable can take. In this article, we will focus on solving the inequality $-\frac{3}{4} x + 2 \leq -7$. This inequality involves a linear expression and a constant, and we will use algebraic techniques to isolate the variable and find the solution.

Understanding the Inequality

The given inequality is $-\frac{3}{4} x + 2 \leq -7$. To solve this inequality, we need to isolate the variable $x$. The first step is to subtract $2$ from both sides of the inequality, which gives us $-\frac{3}{4} x \leq -9$.

Isolating the Variable

To isolate the variable $x$, we need to get rid of the coefficient $-\frac{3}{4}$ that is being multiplied by $x$. We can do this by multiplying both sides of the inequality by the reciprocal of the coefficient, which is $-\frac{4}{3}$. However, when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

Solving the Inequality

Multiplying both sides of the inequality $-\frac{3}{4} x \leq -9$ by $-\frac{4}{3}$ gives us $x \geq 12$. Therefore, the solution to the inequality $-\frac{3}{4} x + 2 \leq -7$ is $x \geq 12$.

Conclusion

In this article, we solved the inequality $-\frac{3}{4} x + 2 \leq -7$ by isolating the variable $x$ and finding the range of values that it can take. The solution to the inequality is $x \geq 12$, which means that the value of $x$ must be greater than or equal to $12$.

Answer

The correct answer is B. $x \geq 12$.

Step-by-Step Solution

Here are the step-by-step solutions to the inequality:

1. Subtract $2$ from both sides of the inequality: $-\frac{3}{4} x + 2 \leq -7 \Rightarrow -\frac{3}{4} x \leq -9$
2. Multiply both sides of the inequality by $-\frac{4}{3}$: $-\frac{3}{4} x \leq -9 \Rightarrow x \geq 12$

Frequently Asked Questions

• What is the solution to the inequality $-\frac{3}{4} x + 2 \leq -7$?
• How do I isolate the variable in an inequality?
• What is the difference between multiplying and dividing both sides of an inequality by a negative number?

Final Answer

The final answer is B. $x \geq 12$.

Introduction

Solving inequalities is a crucial aspect of algebra and is used to find the range of values that a variable can take. In this article, we will provide answers to frequently asked questions related to solving inequalities, including how to isolate the variable, how to handle negative coefficients, and how to determine the direction of the inequality sign.

Q&A

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to isolate the variable. This can be done by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides by the same non-zero value.

Q: How do I handle negative coefficients when solving an inequality?

A: When solving an inequality, if you multiply or divide both sides by a negative number, you need to reverse the direction of the inequality sign. For example, if you have the inequality $x \leq 5$ and you multiply both sides by $-1$, the inequality becomes $x \geq -5$.

Q: What is the difference between multiplying and dividing both sides of an inequality by a negative number?

A: When you multiply or divide both sides of an inequality by a negative number, you need to reverse the direction of the inequality sign. For example, if you have the inequality $x \leq 5$ and you multiply both sides by $-1$, the inequality becomes $x \geq -5$. However, if you divide both sides by $-1$, the inequality becomes $x \geq 5$.

Q: How do I determine the direction of the inequality sign?

A: The direction of the inequality sign depends on the operation you are performing. If you are adding or subtracting a value from both sides of the inequality, the direction of the inequality sign remains the same. However, if you are multiplying or dividing both sides of the inequality by a negative number, you need to reverse the direction of the inequality sign.

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

A: To solve the inequality $x + 2 \leq 5$, we need to isolate the variable $x$. We can do this by subtracting $2$ from both sides of the inequality, which gives us $x \leq 3$.

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

A: To solve the inequality $x - 3 \geq 2$, we need to isolate the variable $x$. We can do this by adding $3$ to both sides of the inequality, which gives us $x \geq 5$.

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

A: To solve the inequality $-\frac{1}{2} x + 3 \leq 2$, we need to isolate the variable $x$. We can do this by subtracting $3$ from both sides of the inequality, which gives us $-\frac{1}{2} x \leq -1$. Then, we can multiply both sides of the inequality by $-2$, which gives us $x \geq 2$.

Conclusion

Solving inequalities is a crucial aspect of algebra and is used to find the range of values that a variable can take. In this article, we provided answers to frequently asked questions related to solving inequalities, including how to isolate the variable, how to handle negative coefficients, and how to determine the direction of the inequality sign.

Final Answer

The final answer is that solving inequalities requires a clear understanding of algebraic operations and the ability to isolate the variable. By following the steps outlined in this article, you can solve inequalities and determine the range of values that a variable can take.