Use The Inequality $18 \ \textless \ -3(4x - 2$\].a. Solve The Inequality For $x$.b. Which Graph Shows The Solution To The Inequality?A. B. C. D.

Introduction

In this article, we will delve into the world of linear inequalities and explore how to solve them. Linear inequalities are mathematical expressions that contain a variable and a constant, connected by a comparison operator such as greater than or less than. In this case, we will be working with the inequality $18 \ \textless \ -3(4x - 2)$. Our goal is to solve this inequality for the variable $x$ and determine which graph represents the solution.

Step 1: Distribute the Negative 3

To begin solving the inequality, we need to distribute the negative 3 to the terms inside the parentheses. This will give us:

$$18 \ \textless \ -12x + 6$$

Step 2: Add 12x to Both Sides

Next, we want to isolate the variable $x$ on one side of the inequality. To do this, we will add $12x$ to both sides of the inequality. This will give us:

$$18 + 12x \ \textless \ 6$$

Step 3: Subtract 18 from Both Sides

Now, we want to get rid of the constant term on the left-hand side of the inequality. To do this, we will subtract 18 from both sides of the inequality. This will give us:

$$12x \ \textless \ -12$$

Step 4: Divide Both Sides by 12

Finally, we want to solve for $x$ by dividing both sides of the inequality by 12. This will give us:

$$x \ \textless \ -1$$

Conclusion

In conclusion, we have successfully solved the inequality $18 \ \textless \ -3(4x - 2)$ for the variable $x$. The solution to the inequality is $x \ \textless \ -1$. Now, let's move on to the next part of the problem, which is to determine which graph represents the solution to the inequality.

Which Graph Shows the Solution to the Inequality?

To determine which graph represents the solution to the inequality, we need to recall that the solution to the inequality $x \ \textless \ -1$ is all values of $x$ that are less than -1. This means that the graph should show all values of $x$ that are less than -1.

Graph A

Graph A shows a line at $x = -1$ and shading to the left of the line. This graph represents the solution to the inequality $x \ \textless \ -1$.

Graph B

Graph B shows a line at $x = -1$ and shading to the right of the line. This graph does not represent the solution to the inequality $x \ \textless \ -1$.

Graph C

Graph C shows a line at $x = -1$ and shading to the left and right of the line. This graph does not represent the solution to the inequality $x \ \textless \ -1$.

Graph D

Graph D shows a line at $x = -1$ and no shading. This graph does not represent the solution to the inequality $x \ \textless \ -1$.

Answer

The graph that shows the solution to the inequality is Graph A.

Discussion

In this article, we have learned how to solve linear inequalities and determine which graph represents the solution. We have also seen how to use the solution to the inequality to determine which graph represents the solution. This is an important skill to have in mathematics, as it allows us to visualize and understand the solution to an inequality.

Key Takeaways

• To solve a linear inequality, we need to isolate the variable on one side of the inequality.
• We can use addition, subtraction, multiplication, and division to isolate the variable.
• The solution to an inequality is all values of the variable that satisfy the inequality.
• We can use graphs to visualize and understand the solution to an inequality.

Conclusion

Q: What is a linear inequality?

A: A linear inequality is a mathematical expression that contains a variable and a constant, connected by a comparison operator such as greater than or less than.

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. You can use addition, subtraction, multiplication, and division to isolate the variable.

Q: What is the first step in solving a linear inequality?

A: The first step in solving a linear inequality is to distribute any negative numbers to the terms inside the parentheses.

Q: How do I know which direction to shade the graph?

A: To determine which direction to shade the graph, you need to look at the inequality sign. If the inequality sign is less than (<), you should shade to the left of the line. If the inequality sign is greater than (>), you should shade to the right of the line.

Q: Can I use the same steps to solve a compound inequality?

A: No, you cannot use the same steps to solve a compound inequality. A compound inequality is an inequality that contains two or more inequalities separated by the word "and" or "or". To solve a compound inequality, you need to use a different set of steps.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each inequality separately and then combine the solutions. If the compound inequality is in the form of "and", you need to find the intersection of the two solutions. If the compound inequality is in the form of "or", you need to find the union of the two solutions.

Q: Can I use a graph to solve a linear inequality?

A: Yes, you can use a graph to solve a linear inequality. A graph can help you visualize the solution to the inequality and determine which direction to shade the graph.

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

A: A linear inequality is an inequality that contains a linear expression, while a quadratic inequality is an inequality that contains a quadratic expression. Quadratic inequalities are more complex and require a different set of steps to solve.

Q: Can I use the same steps to solve a quadratic inequality?

A: No, you cannot use the same steps to solve a quadratic inequality. Quadratic inequalities require a different set of steps, including factoring, using the quadratic formula, or graphing.

Q: How do I know if a quadratic inequality is in the form of "and" or "or"?

A: To determine if a quadratic inequality is in the form of "and" or "or", you need to look at the inequality sign. If the inequality sign is less than (<) or greater than (>), the inequality is in the form of "and". If the inequality sign is less than or equal to (≤) or greater than or equal to (≥), the inequality is in the form of "or".

Q: Can I use a graph to solve a quadratic inequality?

A: Yes, you can use a graph to solve a quadratic inequality. A graph can help you visualize the solution to the inequality and determine which direction to shade the graph.

Conclusion

In conclusion, solving linear inequalities is an important skill to have in mathematics. By following the steps outlined in this article, you can solve linear inequalities and determine which graph represents the solution. Remember to use the same steps to solve a compound inequality, and to use a graph to visualize the solution to an inequality.