What is the solution to $-2(8 x-4)<2 x+5$?
A. $x>\frac{1}{6}$
B. $x<\frac{1}{6}$
C. $x>6$
D. $x<6$
Answer (1)
Distribute -2 on the left side: − 16 x + 8 < 2 x + 5 .
Add 16 x to both sides: 8 < 18 x + 5 .
Subtract 5 from both sides: 3 < 18 x .
Divide both sides by 18: \frac{1}{6}"> x > 6 1 .
Explanation
Understanding the Inequality We are given the inequality − 2 ( 8 x − 4 ) < 2 x + 5 . Our goal is to isolate x on one side of the inequality to find the solution.
Distributing the -2 First, distribute the − 2 on the left side of the inequality: − 2 ( 8 x − 4 ) = − 16 x + 8 So the inequality becomes: − 16 x + 8 < 2 x + 5
Adding 16x to Both Sides Next, we want to isolate the x terms. Add 16 x to both sides of the inequality: − 16 x + 8 + 16 x < 2 x + 5 + 16 x 8 < 18 x + 5
Subtracting 5 from Both Sides Now, subtract 5 from both sides of the inequality: 8 − 5 < 18 x + 5 − 5 3 < 18 x
Dividing by 18 To solve for x , divide both sides of the inequality by 18 :
18 3 < 18 18 x 6 1 < x
Final Solution This can also be written as: \frac{1}{6}"> x > 6 1
Examples
Imagine you're trying to determine the minimum number of items you need to sell to make a profit. If your profit is modeled by the inequality − 2 ( 8 x − 4 ) < 2 x + 5 , where x represents the number of items, solving this inequality tells you the minimum number of items ( x ) you must sell to ensure your profit stays above a certain level. Understanding and solving such inequalities is crucial in business and economics for making informed decisions about production, sales, and investments. In this case, you need to sell more than 6 1 of an item, which practically means you need to sell at least one item to start making a profit.
