Solve The Inequality: ${ 21 \leq -3(x-4) \ \textless \ 30 }$A. { -28 \leq X \ \textless \ -37$}$B. { -3 \leq X \ \textless \ 6$}$C. { -10 \ \textless \ X \leq -3$} D . \[ D. \[ D . \[ -6 \ \textless \ X \leq
Introduction
Inequalities are mathematical expressions that compare two values, often using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. In this article, we will focus on solving the inequality 21 ≤ -3(x-4) < 30, and explore the different solution options.
Understanding the Inequality
The given inequality is 21 ≤ -3(x-4) < 30. To solve this inequality, we need to isolate the variable x. The first step is to distribute the -3 to the terms inside the parentheses.
Distributing the -3
-3(x-4) = -3x + 12
Now, the inequality becomes:
21 ≤ -3x + 12 < 30
Isolating the Variable
To isolate the variable x, we need to get rid of the constant term 12 on the right-hand side of the inequality. We can do this by subtracting 12 from all three parts of the inequality.
21 - 12 ≤ -3x + 12 - 12 < 30 - 12
This simplifies to:
9 ≤ -3x < 18
Dividing by -3
To isolate the variable x, we need to divide all three parts of the inequality by -3. However, when dividing or multiplying an inequality by a negative number, we need to reverse the direction of the inequality signs.
-3(9) ≥ -3x ≥ -3(18)
This simplifies to:
-27 ≥ -3x ≥ -54
Multiplying by -1
To make the inequality easier to read, we can multiply all three parts of the inequality by -1. This will reverse the direction of the inequality signs.
27 ≤ 3x ≤ 54
Dividing by 3
Finally, we can divide all three parts of the inequality by 3 to isolate the variable x.
9 ≤ x ≤ 18
Conclusion
In conclusion, the solution to the inequality 21 ≤ -3(x-4) < 30 is 9 ≤ x ≤ 18. This means that the value of x must be greater than or equal to 9 and less than or equal to 18.
Comparison with Solution Options
Let's compare our solution with the given solution options:
- A. -28 ≤ x < -37: This solution is not correct, as it does not match our solution.
- B. -3 ≤ x < 6: This solution is not correct, as it does not match our solution.
- C. -10 < x ≤ -3: This solution is not correct, as it does not match our solution.
- D. -6 < x ≤ 6: This solution is not correct, as it does not match our solution.
Final Answer
Q&A: Solving Inequalities
Q: What is an inequality?
A: An inequality is a mathematical expression that compares two values, often using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols.
Q: How do I solve an inequality?
A: To solve an inequality, you need to isolate the variable on one side of the inequality sign, while maintaining the direction of the inequality. This involves using inverse operations, such as addition, subtraction, multiplication, and division, to get the variable by itself.
Q: What is the first step in solving an inequality?
A: The first step in solving an inequality is to simplify the expression by combining like terms and removing any parentheses.
Q: How do I handle negative numbers when solving an inequality?
A: When dividing or multiplying an inequality by a negative number, you need to reverse the direction of the inequality signs.
Q: What is the difference between a linear inequality and a quadratic inequality?
A: A linear inequality is an inequality that can be written in the form ax + b ≤ c or ax + b ≥ c, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form ax^2 + bx + c ≤ d or ax^2 + bx + c ≥ d, where a, b, c, and d are constants.
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. If the inequality is of the form x ≤ a or x ≥ a, you need to plot a closed circle at the point a. If the inequality is of the form x < a or x > a, you need to plot an open circle at the point a.
Q: What is the solution to the inequality 21 ≤ -3(x-4) < 30?
A: The solution to the inequality 21 ≤ -3(x-4) < 30 is 9 ≤ x ≤ 18.
Q: How do I check my solution to an inequality?
A: To check your solution to an inequality, you need to plug in a value from the solution set into the original inequality and verify that it is true.
Q: What are some common mistakes to avoid when solving inequalities?
A: Some common mistakes to avoid when solving inequalities include:
- Not reversing the direction of the inequality signs when dividing or multiplying by a negative number
- Not combining like terms and removing parentheses
- Not checking the solution to the inequality
Conclusion
In conclusion, solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. By following the steps outlined in this article, you can solve inequalities and graph them on a number line. Remember to check your solution to an inequality to ensure that it is true.
Additional Resources
For more information on solving inequalities, check out the following resources:
- Khan Academy: Solving Inequalities
- Mathway: Solving Inequalities
- Purplemath: Solving Inequalities
Final Answer
The final answer is 9 ≤ x ≤ 18.