Which value is a solution to the inequality $5x - 1 \leq -11$?
A) $x=-1$
B) $x=0$
C) $x=5$
D) $x=-2$
Answer (1)
Test each option in the inequality 5 x − 1 ≤ − 11 .
For x = − 1 , 5 ( − 1 ) − 1 = − 6 , which is not less than or equal to − 11 .
For x = 0 , 5 ( 0 ) − 1 = − 1 , which is not less than or equal to − 11 .
For x = 5 , 5 ( 5 ) − 1 = 24 , which is not less than or equal to − 11 .
For x = − 2 , 5 ( − 2 ) − 1 = − 11 , which is less than or equal to − 11 . Therefore, the solution is x = − 2 .
Explanation
Understanding the Problem We are given the inequality 5 x − 1 ≤ − 11 , and we need to determine which of the provided options satisfies this inequality. The options are x = − 1 , 0 , 5 , − 2 .
Testing Each Option To find the solution, we can test each value of x in the inequality:
For x = − 1 :
5 ( − 1 ) − 1 = − 5 − 1 = − 6 Since − 6 ≤ − 11 is false, x = − 1 is not a solution.
For x = 0 :
5 ( 0 ) − 1 = 0 − 1 = − 1 Since − 1 ≤ − 11 is false, x = 0 is not a solution.
For x = 5 :
5 ( 5 ) − 1 = 25 − 1 = 24 Since 24 ≤ − 11 is false, x = 5 is not a solution.
For x = − 2 :
5 ( − 2 ) − 1 = − 10 − 1 = − 11 Since − 11 ≤ − 11 is true, x = − 2 is a solution.
Final Answer Therefore, the value that satisfies the inequality 5 x − 1 ≤ − 11 is x = − 2 .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, when budgeting, you might need to ensure that your expenses ( 5 x − 1 ) are less than or equal to your income ( ≤ − 11 ). Solving inequalities helps you determine the maximum amount you can spend while staying within your budget. Similarly, in engineering, inequalities are used to ensure that structures can withstand certain loads or stresses without failing. This problem demonstrates a fundamental concept with wide-ranging applications.
