Two-fifths of one less than a number is less than three-fifths of one more than that number. What numbers are in the solution set of this problem?
A. [tex]$x\ \textless \ -5$[/tex]
B. [tex]$x\ \textgreater \ -5$[/tex]
C. [tex]$x\ \textgreater \ -1$[/tex]
D. [tex]$x\ \textless \ -1$[/tex]
Answer (2)
The inequality set up from the problem results in -5"> x > − 5 . Therefore, the correct choice is B) -5"> x > − 5 .
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Set up the inequality: 5 2 ( x − 1 ) < 5 3 ( x + 1 ) .
Multiply both sides by 5: 2 ( x − 1 ) < 3 ( x + 1 ) .
Expand both sides: 2 x − 2 < 3 x + 3 .
Solve for x : -5"> x > − 5 . The solution set is -5}"> x > − 5 .
Explanation
Understanding the Problem Let's break down the problem step by step. We need to translate the word problem into a mathematical inequality and then solve for the unknown number.
Setting up the Inequality Let x be the number we are trying to find. The problem states: "Two-fifths of one less than a number is less than three-fifths of one more than that number." We can write this as an inequality: 5 2 ( x − 1 ) < 5 3 ( x + 1 )
Eliminating Fractions To solve the inequality, our first goal is to eliminate the fractions. We can do this by multiplying both sides of the inequality by 5: 5 × 5 2 ( x − 1 ) < 5 × 5 3 ( x + 1 ) This simplifies to: 2 ( x − 1 ) < 3 ( x + 1 )
Expanding the Inequality Next, we expand both sides of the inequality by distributing the numbers outside the parentheses: 2 x − 2 < 3 x + 3
Isolating x Now, we want to isolate x on one side of the inequality. Let's subtract 2 x from both sides: 2 x − 2 − 2 x < 3 x + 3 − 2 x This simplifies to: − 2 < x + 3
Solving for x Finally, we subtract 3 from both sides to solve for x : − 2 − 3 < x + 3 − 3 This gives us: − 5 < x So, -5"> x > − 5 . This means that any number greater than -5 satisfies the given condition.
Final Answer Therefore, the solution set for this problem is all numbers greater than -5.
Examples
Imagine you're managing a small business and want to ensure your profits are higher than your costs. This problem is similar to calculating the break-even point, where you need to find the minimum number of sales ( x ) to make sure your revenue is greater than your expenses. By setting up an inequality, you can determine the minimum sales required to achieve profitability. For example, if your profit per sale is represented by 5 2 ( x − 1 ) and your costs are represented by 5 3 ( x + 1 ) , solving the inequality 5 2 ( x − 1 ) < 5 3 ( x + 1 ) will tell you the minimum number of sales needed to avoid losses.
