Solve the following inequality for $x$.
$-8(x-8)<72$
Simplify your answer completely.
Graph the solution
Answer (2)
Distribute -8: − 8 x + 64 < 72 .
Subtract 64 from both sides: − 8 x < 8 .
Divide by -8 and flip the inequality: -1"> x > − 1 .
The solution is -1"> x > − 1 , which is graphed on a number line with an open circle at -1 and shading to the right. -1}"> x > − 1
Explanation
Understanding the Problem We are given the inequality − 8 ( x − 8 ) < 72 . Our goal is to solve for x and simplify the solution completely. We will also graph the solution on a number line.
Distributing the -8 First, we distribute the − 8 on the left side of the inequality: − 8 ( x − 8 ) < 72
− 8 x + 64 < 72
Subtracting 64 from Both Sides Next, we subtract 64 from both sides of the inequality: − 8 x + 64 − 64 < 72 − 64
− 8 x < 8
Dividing by -8 and Flipping the Inequality Now, we divide both sides of the inequality by − 8 . Remember that when we divide by a negative number, we must flip the inequality sign: \frac{8}{-8}"> − 8 − 8 x > − 8 8
-1"> x > − 1
Solution to the Inequality The solution to the inequality is -1"> x > − 1 . This means that x can be any number greater than − 1 .
Graphing the Solution To graph the solution, we draw a number line. We place an open circle at − 1 on the number line because the inequality is strict (i.e., x is strictly greater than − 1 , so − 1 is not included in the solution). Then, we shade the region to the right of − 1 , indicating that all values greater than − 1 are solutions to the inequality.
Examples
Imagine you're setting a minimum score to pass a level in a game. If the inequality is -1"> x > − 1 , it means you need to score higher than -1 to pass. This concept applies to various real-life scenarios, such as setting minimum requirements for age, height, or test scores. Understanding inequalities helps in establishing boundaries and conditions in everyday situations.
To solve the inequality − 8 ( x − 8 ) < 72 , we distribute, simplify, and divide while flipping the inequality, resulting in -1"> x > − 1 . This can be graphed with an open circle at -1 and shading to the right. The final solution is -1"> x > − 1 .
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