What Is The Solution To $-2(8x - 4) \ \textless \ 2x + 5$?A. $x \ \textgreater \ \frac{1}{6}$ B. $x \ \textless \ \frac{1}{6}$ C. $x \ \textgreater \ 6$ D. $x \ \textless \ 6$

Solving Inequalities: A Step-by-Step Guide to Understanding the Solution to $-2(8x - 4) \ \textless \ 2x + 5$

Introduction to Inequalities

Inequalities are mathematical expressions that compare two values, indicating whether one value is greater than, less than, or equal to another value. In this article, we will focus on solving the inequality $-2(8x - 4) \ \textless \ 2x + 5$. To solve this inequality, we will use the properties of inequalities and algebraic manipulations to isolate the variable x.

Understanding the Given Inequality

The given inequality is $-2(8x - 4) \ \textless \ 2x + 5$. To begin solving this inequality, we need to simplify the left-hand side by distributing the negative 2 to the terms inside the parentheses.

Simplifying the Left-Hand Side

Using the distributive property, we can simplify the left-hand side of the inequality as follows:

$$-2(8x - 4) = -16x + 8$$

So, the inequality becomes:

$$-16x + 8 \ \textless \ 2x + 5$$

Moving All Terms to One Side

To isolate the variable x, we need to move all the terms to one side of the inequality. We can do this by subtracting 2x from both sides and adding 16x to both sides.

$$-16x + 8 - 2x \ \textless \ 2x + 5 - 2x$$

This simplifies to:

$$-18x + 8 \ \textless \ 5$$

Simplifying Further

To simplify the inequality further, we can subtract 8 from both sides.

$$-18x + 8 - 8 \ \textless \ 5 - 8$$

This simplifies to:

$$-18x \ \textless \ -3$$

Dividing Both Sides

To isolate the variable x, we need to divide both sides of the inequality by -18. However, when we divide or multiply an inequality by a negative number, we need to reverse the direction of the inequality sign.

$$\frac{-18x}{-18} \ \textgreater \ \frac{-3}{-18}$$

This simplifies to:

$$x \ \textgreater \ \frac{1}{6}$$

Conclusion

In this article, we solved the inequality $-2(8x - 4) \ \textless \ 2x + 5$ by simplifying the left-hand side, moving all terms to one side, and isolating the variable x. The solution to the inequality is $x \ \textgreater \ \frac{1}{6}$. This means that the value of x must be greater than $\frac{1}{6}$ to satisfy the given inequality.

Answer Key

The correct answer is:

A. $x \ \textgreater \ \frac{1}{6}$
Frequently Asked Questions (FAQs) About Solving Inequalities

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, indicating whether one value is greater than, less than, or equal to another value.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by performing algebraic manipulations, such as adding or subtracting the same value to both sides, and multiplying or dividing both sides by the same non-zero value.

Q: What is the difference between solving an equation and solving an inequality?

A: Solving an equation involves finding the value of the variable that makes the equation true, whereas solving an inequality involves finding the set of values of the variable that satisfy the inequality.

Q: How do I know which direction to change the inequality sign when multiplying or dividing both sides by a negative number?

A: When multiplying or dividing both sides of an inequality by a negative number, you need to reverse the direction of the inequality sign.

Q: Can I add or subtract the same value to both sides of an inequality?

A: Yes, you can add or subtract the same value to both sides of an inequality, but you need to make sure that the value is not zero.

Q: Can I multiply or divide both sides of an inequality by zero?

A: No, you cannot multiply or divide both sides of an inequality by zero, as this would result in an undefined expression.

Q: How do I know if an inequality is true or false?

A: To determine if an inequality is true or false, you need to test the inequality by substituting a value of the variable into the inequality and checking if the inequality holds true.

Q: Can I use the same methods to solve linear inequalities as I would to solve linear equations?

A: Yes, you can use the same methods to solve linear inequalities as you would to solve linear equations, but you need to be careful when multiplying or dividing both sides by a negative number.

Q: What is the solution to the inequality $-2(8x - 4) \ \textless \ 2x + 5$?

A: The solution to the inequality $-2(8x - 4) \ \textless \ 2x + 5$ is $x \ \textgreater \ \frac{1}{6}$.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to plot a point on the number line that represents the solution to the inequality and then shade the region to the left or right of the point, depending on the direction of the inequality sign.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality, but you need to make sure that the calculator is set to the correct mode and that you are using the correct operations.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you need to substitute the solution into the original inequality and check if the inequality holds true.