Solve the inequality.
$16\left(\frac{1}{4} x-\frac{1}{2}\right)>24-2 x$
A. $x>-\frac{10}{3}$
B. $x>6$
C. $x>\frac{16}{3}$
D. $x>-6$
Answer (1)
Distribute 16 on the left side: 24 - 2x"> 4 x − 8 > 24 − 2 x .
Add 2 x to both sides: 24"> 6 x − 8 > 24 .
Add 8 to both sides: 32"> 6 x > 32 .
Divide by 6: \frac{16}{3}"> x > 3 16 .
\frac{16}{3}}"> x > 3 16
Explanation
Understanding the Inequality We are given the inequality 24-2 x"> 16 ( 4 1 x − 2 1 ) > 24 − 2 x . Our goal is to isolate x on one side of the inequality to find the solution.
Distributing and Simplifying First, distribute the 16 on the left side of the inequality: 16 ( 4 1 x − 2 1 ) = 16 × 4 1 x − 16 × 2 1 = 4 x − 8 So the inequality becomes: 24 - 2x"> 4 x − 8 > 24 − 2 x
Combining x Terms Next, add 2 x to both sides of the inequality to get all the x terms on one side: 24 - 2x + 2x"> 4 x − 8 + 2 x > 24 − 2 x + 2 x 24"> 6 x − 8 > 24
Isolating the x Term Now, add 8 to both sides of the inequality to isolate the x term: 24 + 8"> 6 x − 8 + 8 > 24 + 8 32"> 6 x > 32
Solving for x Finally, divide both sides of the inequality by 6 to solve for x : \frac{32}{6}"> 6 6 x > 6 32 \frac{32}{6}"> x > 6 32 Simplify the fraction: 6 32 = 3 16 So the solution is: \frac{16}{3}"> x > 3 16
Final Answer The solution to the inequality is \frac{16}{3}"> x > 3 16 . Therefore, the correct answer is C.
Examples
Imagine you're saving money, and you want to know how many weeks you need to save to have more than a certain amount. This inequality is similar to figuring out how many weeks ( x ) you need to save, given that you save a certain amount each week and have some initial expenses. Solving the inequality tells you the minimum number of weeks you need to save to reach your goal. Understanding inequalities helps in managing budgets and planning savings.
