Solve The Inequality: ${ 9|5 - 9x| + 3 \geq -33 }$
Introduction
In this article, we will delve into solving the given inequality, which involves absolute value. The inequality is 9|5 - 9x| + 3 ≥ -33. We will break down the solution step by step, using algebraic manipulations and properties of absolute value. Our goal is to isolate the variable x and find the solution set that satisfies the given inequality.
Understanding Absolute Value
Before we dive into solving the inequality, let's recall the definition of absolute value. The absolute value of a number a, denoted by |a|, is the distance of a from zero on the number line. It is always non-negative, and it can be defined as:
|a| = a if a ≥ 0 |a| = -a if a < 0
Step 1: Isolate the Absolute Value Expression
To solve the inequality, we first need to isolate the absolute value expression. We can do this by subtracting 3 from both sides of the inequality:
9|5 - 9x| ≥ -33 - 3 9|5 - 9x| ≥ -36
Step 2: Divide by 9
Next, we can divide both sides of the inequality by 9 to get:
|5 - 9x| ≥ -36/9 |5 - 9x| ≥ -4
Step 3: Remove the Absolute Value
To remove the absolute value, we need to consider two cases: when the expression inside the absolute value is non-negative, and when it is negative.
Case 1: 5 - 9x ≥ 0
In this case, we can remove the absolute value by simply removing the bars:
5 - 9x ≥ -4
Case 2: 5 - 9x < 0
In this case, we need to remove the absolute value by changing the sign of the expression inside the bars:
-(5 - 9x) ≥ -4
Step 4: Solve the Inequality
Now, we can solve the inequality in each case.
Case 1: 5 - 9x ≥ -4
We can add 9x to both sides of the inequality to get:
5 ≥ -4 + 9x
Subtracting 5 from both sides gives:
0 ≥ -9 + 9x
Adding 9 to both sides gives:
9 ≥ 9x
Dividing both sides by 9 gives:
1 ≥ x
Case 2: -(5 - 9x) ≥ -4
We can simplify the inequality by multiplying both sides by -1:
-(5 - 9x) ≥ -4 -(5 - 9x) + 4 ≥ 0 -5 + 9x + 4 ≥ 0 9x - 1 ≥ 0
Adding 1 to both sides gives:
9x ≥ 1
Dividing both sides by 9 gives:
x ≥ 1/9
Conclusion
In conclusion, we have solved the given inequality by isolating the absolute value expression, removing the absolute value, and solving the resulting inequality in each case. The solution set is x ≥ 1/9 or x ≤ 1.
Final Answer
The final answer is x ∈ [1/9, 1].
Introduction
In our previous article, we solved the inequality 9|5 - 9x| + 3 ≥ -33 by isolating the absolute value expression, removing the absolute value, and solving the resulting inequality in each case. The solution set is x ≥ 1/9 or x ≤ 1. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.
Q&A
Q: What is the definition of absolute value?
A: The absolute value of a number a, denoted by |a|, is the distance of a from zero on the number line. It is always non-negative, and it can be defined as:
|a| = a if a ≥ 0 |a| = -a if a < 0
Q: How do I isolate the absolute value expression in an inequality?
A: To isolate the absolute value expression, you need to get the absolute value term by itself on one side of the inequality. You can do this by subtracting or adding the same value to both sides of the inequality.
Q: What is the difference between the two cases when removing the absolute value?
A: When removing the absolute value, you need to consider two cases: when the expression inside the absolute value is non-negative, and when it is negative. In the first case, you can simply remove the absolute value by removing the bars. In the second case, you need to remove the absolute value by changing the sign of the expression inside the bars.
Q: How do I solve the inequality in each case?
A: To solve the inequality in each case, you need to follow the same steps as solving a linear inequality. You can add or subtract the same value to both sides of the inequality, and then divide both sides by a non-zero value.
Q: What is the solution set for the inequality 9|5 - 9x| + 3 ≥ -33?
A: The solution set for the inequality 9|5 - 9x| + 3 ≥ -33 is x ≥ 1/9 or x ≤ 1.
Q: Can I use a calculator to solve the inequality?
A: Yes, you can use a calculator to solve the inequality. However, 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?
A: To check your solution, you need to plug the solution back into the original inequality and make sure that it is true. If the solution is not true, then you need to recheck your work and try again.
Conclusion
In conclusion, we have provided a Q&A section to help clarify any doubts or questions that readers may have about solving the inequality 9|5 - 9x| + 3 ≥ -33. We hope that this Q&A section has been helpful in understanding the solution to the inequality.
Final Answer
The final answer is x ∈ [1/9, 1].