$2 \leqslant \frac{2}{3}(5-x)$
Answer (1)
Multiply both sides by 2 3 : 3 ⩽ 5 − x .
Subtract 5 from both sides: − 2 ⩽ − x .
Multiply by -1 (and flip the inequality sign): 2 ⩾ x .
The solution is x ⩽ 2 , meaning x is less than or equal to 2. x ⩽ 2
Explanation
Understanding the Problem We are given the inequality 2 l e q s l an t 3 2 ( 5 − x ) and our goal is to solve for x . This means we want to isolate x on one side of the inequality to find the range of values that satisfy the given condition.
Eliminating the Fraction To begin, let's multiply both sides of the inequality by 2 3 to eliminate the fraction on the right side: 2 3 ⋅ 2 ⩽ 2 3 ⋅ 3 2 ( 5 − x ) This simplifies to: 3 ⩽ 5 − x
Isolating x Next, we want to isolate x . We can subtract 5 from both sides of the inequality: 3 − 5 ⩽ 5 − x − 5 This simplifies to: − 2 ⩽ − x
Solving for x Now, we need to get rid of the negative sign in front of x . We can multiply both sides of the inequality by -1. Remember that when we multiply or divide an inequality by a negative number, we must flip the direction of the inequality sign: ( − 1 ) ⋅ ( − 2 ) ⩾ ( − 1 ) ⋅ ( − x ) This simplifies to: 2 ⩾ x
Final Answer This inequality can also be written as: x ⩽ 2 This means that x is less than or equal to 2.
Examples
Imagine you're baking a cake and the recipe says you need at least 2 cups of flour. If you have a measuring cup that only fills two-thirds at a time, and you're trying to figure out how much flour is left in a 5-cup bag after using some, this inequality helps you determine the maximum amount you could have used so that you still have enough for the recipe. Understanding inequalities is crucial in managing resources and ensuring you meet minimum requirements in various real-life situations.
