Select The Correct Answer.Which Graph Represents The Solution To This Inequality? − 1 4 ( 12 X + 8 ) ≤ − 2 X + 11 -\frac{1}{4}(12x + 8) \leq -2x + 11 − 4 1 ​ ( 12 X + 8 ) ≤ − 2 X + 11 A. B. C. D.

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Introduction

In mathematics, inequalities are a fundamental concept that helps us compare the values of different expressions. Solving linear inequalities involves finding the solution set that satisfies the given inequality. In this article, we will focus on solving the inequality −14(12x+8)≤−2x+11-\frac{1}{4}(12x + 8) \leq -2x + 11 and determine which graph represents the solution to this inequality.

Understanding the Inequality

The given inequality is −14(12x+8)≤−2x+11-\frac{1}{4}(12x + 8) \leq -2x + 11. To solve this inequality, we need to isolate the variable xx on one side of the inequality sign. We can start by simplifying the left-hand side of the inequality.

Simplifying the Left-Hand Side

To simplify the left-hand side, we can distribute the negative 14\frac{1}{4} to the terms inside the parentheses:

−14(12x+8)=−3x−2-\frac{1}{4}(12x + 8) = -3x - 2

So, the inequality becomes:

−3x−2≤−2x+11-3x - 2 \leq -2x + 11

Isolating the Variable

Next, we need to isolate the variable xx on one side of the inequality sign. We can do this by adding 3x3x to both sides of the inequality:

−2≤x+11-2 \leq x + 11

Simplifying the Right-Hand Side

Now, we can simplify the right-hand side by subtracting 1111 from both sides of the inequality:

−13≤x-13 \leq x

Graphing the Solution

The solution to the inequality −13≤x-13 \leq x is a closed interval on the number line. The graph of this solution is a closed circle at x=−13x = -13 and an open circle at x=∞x = \infty.

Determining the Correct Graph

Now that we have the solution to the inequality, we can determine which graph represents the solution. The correct graph is the one that shows a closed circle at x=−13x = -13 and an open circle at x=∞x = \infty.

Conclusion

In conclusion, solving linear inequalities involves finding the solution set that satisfies the given inequality. By simplifying the left-hand side, isolating the variable, and graphing the solution, we can determine which graph represents the solution to the inequality −14(12x+8)≤−2x+11-\frac{1}{4}(12x + 8) \leq -2x + 11.

Step-by-Step Solution

Here is a step-by-step solution to the inequality:

  1. Simplify the left-hand side: −14(12x+8)=−3x−2-\frac{1}{4}(12x + 8) = -3x - 2
  2. Isolate the variable: −3x−2≤−2x+11-3x - 2 \leq -2x + 11
  3. Add 3x3x to both sides: −2≤x+11-2 \leq x + 11
  4. Subtract 1111 from both sides: −13≤x-13 \leq x

Graphing the Solution

The graph of the solution is a closed circle at x=−13x = -13 and an open circle at x=∞x = \infty.

Determining the Correct Graph

The correct graph is the one that shows a closed circle at x=−13x = -13 and an open circle at x=∞x = \infty.

Conclusion

In conclusion, solving linear inequalities involves finding the solution set that satisfies the given inequality. By simplifying the left-hand side, isolating the variable, and graphing the solution, we can determine which graph represents the solution to the inequality −14(12x+8)≤−2x+11-\frac{1}{4}(12x + 8) \leq -2x + 11.

Final Answer

The final answer is:

A.

Note: The final answer is based on the assumption that the correct graph is the one that shows a closed circle at x=−13x = -13 and an open circle at x=∞x = \infty.

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Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression, which is an expression that can be written in the form ax+bax + b, where aa and bb are constants and xx is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. 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: What is the difference between a linear inequality and a linear equation?

A: A linear equation is an equation that involves a linear expression, and it is equal to a constant. A linear inequality, on the other hand, is an inequality that involves a linear expression, and it is not equal to a constant.

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 number line and mark the values that satisfy the inequality. If the inequality is of the form ax+b≤cax + b \leq c, you will draw a closed circle at the value of cc. If the inequality is of the form ax+b≥cax + b \geq c, you will draw an open circle at the value of cc.

Q: What is the solution to the inequality −14(12x+8)≤−2x+11-\frac{1}{4}(12x + 8) \leq -2x + 11?

A: The solution to the inequality −14(12x+8)≤−2x+11-\frac{1}{4}(12x + 8) \leq -2x + 11 is x≥−13x \geq -13.

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 inequality and determine the type of inequality it is. If the inequality is of the form ax+b≤cax + b \leq c, you will look for a closed circle at the value of cc. If the inequality is of the form ax+b≥cax + b \geq c, you will look for an open circle at the value of cc.

Q: What is the final answer to the inequality −14(12x+8)≤−2x+11-\frac{1}{4}(12x + 8) \leq -2x + 11?

A: The final answer to the inequality −14(12x+8)≤−2x+11-\frac{1}{4}(12x + 8) \leq -2x + 11 is:

A.

Note: The final answer is based on the assumption that the correct graph is the one that shows a closed circle at x=−13x = -13 and an open circle at x=∞x = \infty.

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