Solve the following inequality for $x$.
$-8(x-6)>72$
Simplify your answer completely.
Answer (1)
Divide both sides of the inequality by − 8 , flipping the inequality sign: x − 6 < − 9 .
Add 6 to both sides: x < − 3 .
The solution to the inequality is x < − 3 .
Any value of x less than − 3 satisfies the inequality: x < − 3 .
Explanation
Understanding the Problem We are given the inequality 72"> − 8 ( x − 6 ) > 72 . Our goal is to isolate x on one side of the inequality to find the solution.
Dividing by -8 First, we divide both sides of the inequality by − 8 . Remember that when we divide or multiply an inequality by a negative number, we must reverse the direction of the inequality sign. So, we have: − 8 − 8 ( x − 6 ) < − 8 72 x − 6 < − 9
Isolating x Next, we add 6 to both sides of the inequality to isolate x :
x − 6 + 6 < − 9 + 6 x < − 3
Final Answer Therefore, the solution to the inequality is x < − 3 . This means that any value of x that is less than − 3 will satisfy the original inequality.
Examples
Imagine you're managing a budget and need to ensure your expenses don't exceed a certain limit. This inequality helps you determine the maximum amount you can spend while staying within your budget. For example, if you have a budget of $72 and each expense costs $8 less than a certain amount (x), this inequality helps you find the maximum value of x to stay within your budget.
