Solve the inequality for $x$.
$5-\frac{3}{2} x \geq \frac{1}{3}$
A. $x \leq \frac{28}{9}$
B. $x \leq 7$
C. $x \geq \frac{28}{9}$
D. $x \geq 7$
Answer (1)
Subtract 5 from both sides: − 2 3 x ≥ 3 1 − 5 .
Simplify the right side: − 2 3 x ≥ − 3 14 .
Multiply both sides by − 3 2 and flip the inequality sign: x ≤ ( − 3 2 ) ( − 3 14 ) .
Simplify to find the solution: x ≤ 9 28 .
Explanation
Understanding the Problem We are given the inequality 5 − 2 3 x ≥ 3 1 . Our goal is to solve for x .
Isolating the x term First, subtract 5 from both sides of the inequality: 5 − 2 3 x 5 − 2 3 x − 5 − 2 3 x ≥ 3 1 ≥ 3 1 − 5 ≥ 3 1 − 5
Simplifying the Inequality Now, simplify the right side of the inequality. We need to find a common denominator to subtract the fractions. Since 5 = 3 15 , we have: − 2 3 x − 2 3 x ≥ 3 1 − 3 15 ≥ − 3 14
Solving for x Next, we want to isolate x . To do this, we multiply both sides of the inequality by − 3 2 . Remember that when we multiply or divide both sides of an inequality by a negative number, we must flip the inequality sign: − 2 3 x ( − 3 2 ) ( − 2 3 x ) x ≥ − 3 14 ≤ ( − 3 2 ) ( − 3 14 ) ≤ 9 28
Final Answer Therefore, the solution to the inequality is x ≤ 9 28 .
Examples
Imagine you're managing a budget and need to ensure your expenses don't exceed your income. This type of inequality helps you determine the maximum amount you can spend while still staying within your financial limits. For example, if you earn 5 p er h o u r an d n ee d t os a v e a tl e a s t \frac{1}{3}$ of your earnings, the inequality 5 − 2 3 x ≥ 3 1 can help you calculate the maximum number of hours ( x ) you can work on a side project that costs 2 3 per hour, ensuring you still meet your savings goal. Understanding and solving inequalities is crucial for effective financial planning and decision-making.
