Let's break down the problem step-by-step. The problem states: "Two-fifths of one less than a number is less than three-fifths of one more than that number."
1. Let the number be represented by [tex]\( x \)[/tex].
2. According to the problem, we have:
[tex]\[
\frac{2}{5} (x - 1) < \frac{3}{5} (x + 1)
\][/tex]
3. To solve this inequality, let's first eliminate the fractions by multiplying through by 5:
[tex]\[
2(x - 1) < 3(x + 1)
\][/tex]
4. Next, distribute the constants inside the parentheses:
[tex]\[
2x - 2 < 3x + 3
\][/tex]
5. To isolate [tex]\( x \)[/tex], first subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[
-2 < x + 3
\][/tex]
6. Then, subtract 3 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
-2 - 3 < x
\][/tex]
[tex]\[
-5 < x
\][/tex]
Thus, the solution set is:
[tex]\[
x > -5
\][/tex]
In terms of the choices given:
- [tex]\( x < -5 \)[/tex] is incorrect.
- [tex]\( x > -5 \)[/tex] is correct.
- [tex]\( x > -1 \)[/tex] is not correct; it misses all numbers between [tex]\( -5 \)[/tex] and [tex]\( -1 \)[/tex].
- [tex]\( x < -1 \)[/tex] is also incorrect since it does not represent the solution found.
Therefore, the correct inequality is:
[tex]\[
x > -5
\][/tex]
