What Is The Solution To The Inequality?${ 17 \ \textless \ 9 + X }$A. { X \ \textless \ 26 $}$B. { X \ \textgreater \ 26 $}$C. { X \ \textgreater \ 8 $}$D. { X \ \textless \ 8 $}$
Understanding the Inequality
The given inequality is 17 < 9 + x. To solve this inequality, we need to isolate the variable x. The first step is to subtract 9 from both sides of the inequality.
Subtracting 9 from Both Sides
When we subtract 9 from both sides of the inequality, we get:
17 - 9 < 9 + x - 9
This simplifies to:
8 < x
Understanding the Solution
The solution to the inequality is x > 8. This means that any value of x that is greater than 8 will satisfy the inequality.
Analyzing the Options
Now, let's analyze the given options:
A. x < 26 B. x > 26 C. x > 8 D. x < 8
Option A: x < 26
This option is incorrect because the solution to the inequality is x > 8, not x < 26.
Option B: x > 26
This option is also incorrect because the solution to the inequality is x > 8, not x > 26.
Option C: x > 8
This option is correct because it matches the solution to the inequality.
Option D: x < 8
This option is incorrect because the solution to the inequality is x > 8, not x < 8.
Conclusion
The solution to the inequality 17 < 9 + x is x > 8. This means that any value of x that is greater than 8 will satisfy the inequality. Therefore, the correct option is C. x > 8.
Frequently Asked Questions
Q: What is the solution to the inequality 17 < 9 + x?
A: The solution to the inequality is x > 8.
Q: Why is option A incorrect?
A: Option A is incorrect because the solution to the inequality is x > 8, not x < 26.
Q: Why is option B incorrect?
A: Option B is incorrect because the solution to the inequality is x > 8, not x > 26.
Q: Why is option D incorrect?
A: Option D is incorrect because the solution to the inequality is x > 8, not x < 8.
Step-by-Step Solution
- Subtract 9 from both sides of the inequality: 17 - 9 < 9 + x - 9
- Simplify the inequality: 8 < x
- Write the solution in terms of x: x > 8
Common Mistakes
- Subtracting 9 from only one side of the inequality
- Not simplifying the inequality correctly
- Writing the solution in terms of x incorrectly
Real-World Applications
- Inequality solutions are used in a variety of real-world applications, such as finance, economics, and engineering.
- Understanding how to solve inequalities is crucial in making informed decisions and solving complex problems.
Final Thoughts
Solving inequalities is an essential skill in mathematics, and understanding how to solve them can have a significant impact on real-world applications. By following the steps outlined in this article, you can confidently solve inequalities and make informed decisions.
Frequently Asked Questions
Q: What is an inequality?
A: An inequality is a mathematical statement that compares two expressions using a relation such as <, >, ≤, or ≥.
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. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
Q: What is the difference between an inequality and an equation?
A: An equation is a statement that says two expressions are equal, while an inequality is a statement that compares two expressions using a relation such as <, >, ≤, or ≥.
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. This will not change the direction of the inequality.
Q: Can I multiply or divide both sides of an inequality by the same value?
A: Yes, you can multiply or divide both sides of an inequality by the same value, but you need to be careful not to change the direction of the inequality. If you multiply or divide both sides by a negative value, you need to reverse the direction of the inequality.
Q: How do I know when to reverse the direction of an inequality?
A: You need to reverse the direction of an inequality when you multiply or divide both sides by a negative value.
Q: What is the solution to the inequality 2x + 5 > 11?
A: To solve this inequality, you need to isolate the variable x. You can do this by subtracting 5 from both sides of the inequality and then dividing both sides by 2.
2x + 5 - 5 > 11 - 5 2x > 6 x > 3
Q: What is the solution to the inequality x - 3 ≤ 7?
A: To solve this inequality, you need to isolate the variable x. You can do this by adding 3 to both sides of the inequality.
x - 3 + 3 ≤ 7 + 3 x ≤ 10
Q: Can I use the same steps to solve a compound inequality?
A: Yes, you can use the same steps to solve a compound inequality. A compound inequality is an inequality that contains two or more inequalities joined by the word "and" or "or".
Q: How do I solve a compound inequality?
A: To solve a compound inequality, you need to solve each inequality separately and then combine the solutions.
Q: What is the solution to the compound inequality 2x + 5 > 11 and x - 3 ≤ 7?
A: To solve this compound inequality, you need to solve each inequality separately and then combine the solutions.
2x + 5 > 11 2x > 6 x > 3
x - 3 ≤ 7 x ≤ 10
Since x > 3 and x ≤ 10, the solution to the compound inequality is 3 < x ≤ 10.
Common Mistakes
- Not isolating the variable on one side of the inequality sign
- Not reversing the direction of the inequality when multiplying or dividing both sides by a negative value
- Not combining the solutions to a compound inequality correctly
Real-World Applications
- Inequality solutions are used in a variety of real-world applications, such as finance, economics, and engineering.
- Understanding how to solve inequalities is crucial in making informed decisions and solving complex problems.
Final Thoughts
Solving inequalities is an essential skill in mathematics, and understanding how to solve them can have a significant impact on real-world applications. By following the steps outlined in this article, you can confidently solve inequalities and make informed decisions.