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 (1)
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
Setting up the Inequality Let x be the number we are trying to find. The problem states that two-fifths of one less than the 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 this inequality, we first multiply both sides by 5 to eliminate the fractions: 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: 2 x − 2 < 3 x + 3 .
Isolating x Now, we want to isolate x . We can subtract 2 x from both sides: 2 x − 2 − 2 x < 3 x + 3 − 2 x , which simplifies to: − 2 < x + 3 .
Solving for x Finally, we subtract 3 from both sides: − 2 − 3 < x + 3 − 3 , which simplifies to: − 5 < x . This means that x must be greater than -5.
Final Answer Therefore, the solution set for this problem is all numbers greater than -5. In interval notation, this is ( − 5 , ∞ ) . The correct answer from the given options is -5"> x > − 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 setting up a budget where you want two-fifths of your expenses to be less than three-fifths of your income. By solving such inequalities, you can determine the minimum income needed to keep your business profitable. This type of problem helps in making informed financial decisions and planning for sustainable growth.
