We need to solve the following inequality:
[tex]\[ \frac{2}{5}(x - 1) < \frac{3}{5}(x + 1) \][/tex]
We will follow the step-by-step process to solve this inequality.
1. Eliminate the fractions: Multiply both sides of the inequality by 5 to get rid of the denominators:
[tex]\[ 5 \cdot \frac{2}{5}(x - 1) < 5 \cdot \frac{3}{5}(x + 1) \][/tex]
This simplifies to:
[tex]\[ 2(x - 1) < 3(x + 1) \][/tex]
2. Distribute the constants: Use the distributive property to expand both sides:
[tex]\[ 2x - 2 < 3x + 3 \][/tex]
3. Isolate the variable terms: To isolate [tex]\( x \)[/tex] on one side, subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ -2 < x + 3 \][/tex]
4. Isolate the constant terms: Subtract 3 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ -2 - 3 < x \][/tex]
Which simplifies to:
[tex]\[ -5 < x \][/tex]
This means that [tex]\( x \)[/tex] must be greater than -5.
Therefore, the solution set for the inequality is:
[tex]\[ (-5, \infty) \][/tex]
In interval notation, this means [tex]\( x \)[/tex] can be any number greater than -5.
So the correct answer is:
[tex]\[ x > -5 \][/tex]
