Which Are Correct Representations Of The Inequality $-3(2x-5)\ \textless \ 5(2-x$\]?Select Two Options:A. $x\ \textless \ 5$B. $-6x-5\ \textless \ 10-x$C. $-6x+15\ \textless \ 10-5x$

**Solving Inequalities: A Step-by-Step Guide** =====================================================

Understanding the Basics of Inequalities

In mathematics, an inequality is a statement that two expressions are not equal. It can be either greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of a linear equation.

Step 1: Distribute and Simplify

When solving an inequality, the first step is to distribute and simplify the expression. This involves multiplying the numbers outside the parentheses by the terms inside the parentheses.

Example:

Solve the inequality $-3(2x-5) < 5(2-x)$.

Step 2: Distribute the Numbers

Distribute the numbers outside the parentheses to the terms inside the parentheses.

$-3(2x-5) = -6x + 15$

$5(2-x) = 10 - 5x$

Step 3: Simplify the Inequality

Combine like terms and simplify the inequality.

$-6x + 15 < 10 - 5x$

Step 4: Add or Subtract the Same Value

To isolate the variable, add or subtract the same value to both sides of the inequality.

Add $5x$ to both sides:

$-6x + 15 + 5x < 10 - 5x + 5x$

Simplify:

$-x + 15 < 10$

Step 5: Subtract the Same Value

Subtract the same value from both sides of the inequality.

Subtract 15 from both sides:

$-x + 15 - 15 < 10 - 15$

Simplify:

$-x < -5$

Step 6: Multiply or Divide by a Negative Number

When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign changes.

Multiply both sides by -1:

$-x < -5$

Change the direction of the inequality sign:

$x > 5$

Step 7: Write the Solution

The final step is to write the solution to the inequality.

$x > 5$

Conclusion

Solving inequalities requires a step-by-step approach. By distributing and simplifying the expression, adding or subtracting the same value, and multiplying or dividing by a negative number, we can isolate the variable and write the solution to the inequality.

Frequently Asked Questions (FAQs)

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

A: The first step in solving an inequality is to distribute and simplify the expression.

Q: How do I distribute the numbers outside the parentheses?

A: To distribute the numbers outside the parentheses, multiply the numbers by the terms inside the parentheses.

Q: What is the purpose of adding or subtracting the same value to both sides of the inequality?

A: The purpose of adding or subtracting the same value to both sides of the inequality is to isolate the variable.

Q: What happens when multiplying or dividing both sides of an inequality by a negative number?

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

Q: How do I write the solution to an inequality?

A: To write the solution to an inequality, use the correct inequality sign and include the variable and the value it is greater than or less than.

Common Mistakes to Avoid

• Not distributing and simplifying the expression
• Not adding or subtracting the same value to both sides of the inequality
• Not changing the direction of the inequality sign when multiplying or dividing by a negative number
• Not writing the solution in the correct format

Conclusion

Solving inequalities requires a step-by-step approach. By following the steps outlined in this article, you can solve linear inequalities and write the solution in the correct format. Remember to avoid common mistakes and always check your work to ensure accuracy.