Which of the following values are solutions to the inequality $-6-x \leq 7$?
I. -13
II. -14
III. -6
A. None
B. I only
C. II only
D. III only
E. I and II
F. I and III
G. II and III
H. I, II and III
Answer (2)
Solve the inequality − 6 − x ≤ 7 to get xg e q − 13 .
Check if x = − 13 satisfies xg e q − 13 . It does.
Check if x = − 14 satisfies xg e q − 13 . It does not.
Check if x = − 6 satisfies xg e q − 13 . It does. Therefore, the answer is I and III. I an d III
Explanation
Understanding the Problem We are given the inequality − 6 − x ≤ 7 and we need to determine which of the values -13, -14, and -6 are solutions to this inequality. A value is a solution if, when substituted for x , it makes the inequality true.
Solving the Inequality First, let's solve the inequality for x . We have: − 6 − x ≤ 7 Add 6 to both sides: − x ≤ 7 + 6 − x ≤ 13 Multiply both sides by -1. Remember to flip the inequality sign when multiplying by a negative number: xg e q − 13 So, the solution to the inequality is xg e q − 13 .
Checking the Values Now, let's check each of the given values:
I. x = − 13 : Since − 13 g e q − 13 is true, -13 is a solution. II. x = − 14 : Since − 14 g e q − 13 is false, -14 is not a solution. III. x = − 6 : Since − 6 g e q − 13 is true, -6 is a solution.
Conclusion Therefore, the values -13 and -6 are solutions to the inequality.
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, consider a budget constraint: if you have a maximum amount of money to spend, say 100 , an d yo u w an tt o b u y i t e m s t ha t cos t x each, the inequality n x ≤ 100 helps you determine the maximum number n of items you can buy. Similarly, in speed limits, the inequality v ≤ v ma x ensures that your speed v does not exceed the maximum speed limit v ma x , promoting safety and preventing fines. Inequalities are also used in determining the range of acceptable values in manufacturing, ensuring product quality and consistency.
The solutions to the inequality − 6 − x ≤ 7 are -13 and -6. Therefore, options I and III are the correct answers. Thus, the correct multiple choice option is F (I and III).
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