Solve The Inequality:$-10 \ \textless \ 2(4-x) \leq 12$

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve a specific type of inequality. The given inequality is $-10 \ \textless \ 2(4-x) \leq 12$. Our goal is to isolate the variable $x$ and find the range of values that satisfy the given inequality. We will use algebraic manipulations and properties of inequalities to solve this problem.

Understanding the Inequality

The given inequality is a compound inequality, which means it consists of two parts: a less-than inequality and a less-than-or-equal-to inequality. We can rewrite the inequality as $-10 \ \textless \ 2(4-x)$ and $2(4-x) \leq 12$. Our first step is to simplify the expressions inside the parentheses.

Simplifying the Inequality

To simplify the inequality, we will start by evaluating the expression inside the parentheses. We have $2(4-x)$, which can be rewritten as $8-2x$. Now, we can substitute this expression back into the inequality.

Step 1: Simplifying the Less-Than Inequality

We start by simplifying the less-than inequality: $-10 \ \textless \ 8-2x$. To isolate the variable $x$, we will add $2x$ to both sides of the inequality. This gives us $-10+2x \ \textless \ 8$. Next, we will add $10$ to both sides of the inequality to get $2x \ \textless \ 18$. Finally, we will divide both sides of the inequality by $2$ to get $x \ \textless \ 9$.

Step 2: Simplifying the Less-Than-Or-Equal-To Inequality

Now, we will simplify the less-than-or-equal-to inequality: $8-2x \leq 12$. To isolate the variable $x$, we will subtract $8$ from both sides of the inequality. This gives us $-2x \leq 4$. Next, we will divide both sides of the inequality by $-2$. However, when we divide by a negative number, we must reverse the direction of the inequality. This gives us $x \geq -2$.

Combining the Inequalities

Now that we have simplified both inequalities, we can combine them to get the final solution. We have $x \ \textless \ 9$ and $x \geq -2$. To combine these inequalities, we can use the union of intervals. The union of intervals is the set of all values that satisfy at least one of the inequalities.

Finding the Union of Intervals

To find the union of intervals, we can graph the two inequalities on a number line. The first inequality, $x \ \textless \ 9$, is represented by an open circle at $x=9$ and a line extending to the left. The second inequality, $x \geq -2$, is represented by a closed circle at $x=-2$ and a line extending to the right. The union of intervals is the set of all values that lie between the two lines.

Conclusion

In this article, we learned how to solve a compound inequality. We started by simplifying the expressions inside the parentheses and then isolated the variable $x$. We used algebraic manipulations and properties of inequalities to solve the problem. Finally, we combined the two inequalities to get the final solution. The union of intervals is the set of all values that satisfy at least one of the inequalities.

Final Answer

The final answer is $-2 \leq x \ \textless \ 9$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Simplify the expressions inside the parentheses: $2(4-x) = 8-2x$
2. Substitute the expression back into the inequality: $-10 \ \textless \ 8-2x$ and $8-2x \leq 12$
3. Simplify the less-than inequality: $-10+2x \ \textless \ 8$ and $2x \ \textless \ 18$
4. Simplify the less-than-or-equal-to inequality: $-2x \leq 4$ and $x \geq -2$
5. Combine the inequalities: $x \ \textless \ 9$ and $x \geq -2$
6. Find the union of intervals: $-2 \leq x \ \textless \ 9$

Frequently Asked Questions

• Q: What is a compound inequality? A: A compound inequality is an inequality that consists of two parts: a less-than inequality and a less-than-or-equal-to inequality.
• Q: How do I simplify a compound inequality? A: To simplify a compound inequality, you can start by evaluating the expressions inside the parentheses and then isolate the variable.
• Q: What is the union of intervals? A: The union of intervals is the set of all values that satisfy at least one of the inequalities.

Additional Resources

• Khan Academy: Inequalities
• Mathway: Inequalities
• Wolfram Alpha: Inequalities

Conclusion

In this article, we learned how to solve a compound inequality. We started by simplifying the expressions inside the parentheses and then isolated the variable $x$. We used algebraic manipulations and properties of inequalities to solve the problem. Finally, we combined the two inequalities to get the final solution. The union of intervals is the set of all values that satisfy at least one of the inequalities.

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Introduction

In our previous article, we learned how to solve a compound inequality. We started by simplifying the expressions inside the parentheses and then isolated the variable $x$. We used algebraic manipulations and properties of inequalities to solve the problem. Finally, we combined the two inequalities to get the final solution. In this article, we will answer some frequently asked questions about solving compound inequalities.

Q&A

Q: What is a compound inequality?

A: A compound inequality is an inequality that consists of two parts: a less-than inequality and a less-than-or-equal-to inequality.

Q: How do I simplify a compound inequality?

A: To simplify a compound inequality, you can start by evaluating the expressions inside the parentheses and then isolate the variable. You can use algebraic manipulations and properties of inequalities to solve the problem.

Q: What is the union of intervals?

A: The union of intervals is the set of all values that satisfy at least one of the inequalities.

Q: How do I find the union of intervals?

A: To find the union of intervals, you can graph the two inequalities on a number line. The first inequality is represented by an open circle at $x=9$ and a line extending to the left. The second inequality is represented by a closed circle at $x=-2$ and a line extending to the right. The union of intervals is the set of all values that lie between the two lines.

Q: What are some common mistakes to avoid when solving compound inequalities?

A: Some common mistakes to avoid when solving compound inequalities include:

• Not simplifying the expressions inside the parentheses
• Not isolating the variable
• Not using algebraic manipulations and properties of inequalities to solve the problem
• Not combining the two inequalities to get the final solution

Q: How do I check my solution to a compound inequality?

A: To check your solution to a compound inequality, you can plug in a value from the solution set into the original inequality. If the value satisfies the inequality, then your solution is correct.

Q: What are some real-world applications of solving compound inequalities?

A: Solving compound inequalities has many real-world applications, including:

• Finding the range of values for a variable in a mathematical model
• Determining the feasibility of a solution to a problem
• Identifying the constraints on a variable in a system

Conclusion

In this article, we answered some frequently asked questions about solving compound inequalities. We covered topics such as simplifying compound inequalities, finding the union of intervals, and checking solutions. We also discussed some common mistakes to avoid and real-world applications of solving compound inequalities.

Additional Resources

• Khan Academy: Inequalities
• Mathway: Inequalities
• Wolfram Alpha: Inequalities

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Simplify the expressions inside the parentheses: $2(4-x) = 8-2x$
2. Substitute the expression back into the inequality: $-10 \ \textless \ 8-2x$ and $8-2x \leq 12$
3. Simplify the less-than inequality: $-10+2x \ \textless \ 8$ and $2x \ \textless \ 18$
4. Simplify the less-than-or-equal-to inequality: $-2x \leq 4$ and $x \geq -2$
5. Combine the inequalities: $x \ \textless \ 9$ and $x \geq -2$
6. Find the union of intervals: $-2 \leq x \ \textless \ 9$

Frequently Asked Questions

• Q: What is a compound inequality? A: A compound inequality is an inequality that consists of two parts: a less-than inequality and a less-than-or-equal-to inequality.
• Q: How do I simplify a compound inequality? A: To simplify a compound inequality, you can start by evaluating the expressions inside the parentheses and then isolate the variable.
• Q: What is the union of intervals? A: The union of intervals is the set of all values that satisfy at least one of the inequalities.

Additional Resources

• Khan Academy: Inequalities
• Mathway: Inequalities
• Wolfram Alpha: Inequalities

Conclusion

In this article, we learned how to solve a compound inequality. We started by simplifying the expressions inside the parentheses and then isolated the variable $x$. We used algebraic manipulations and properties of inequalities to solve the problem. Finally, we combined the two inequalities to get the final solution. The union of intervals is the set of all values that satisfy at least one of the inequalities.