Inequality (2y+1)/(y+2) < 1 Analysis

The task is to find the range of $y$ for the inequality:

$ \frac{2y+1}{y+2} < 1 $

y-Expression Simplification

Rewrite the inequality with zero on one side:

$ \frac{2y+1}{y+2} - 1 < 0 $

Combine terms using a common denominator $y+2$:

$ \frac{(2y+1) - (y+2)}{y+2} < 0 $

Simplify the numerator:

$ \frac{y - 1}{y+2} < 0 $

Critical Points Determination

Find the values of $y$ where the numerator or denominator is zero:

• Numerator: $y - 1 = 0 \implies y = 1$
• Denominator: $y + 2 = 0 \implies y = -2$

These critical points, $-2$ and $1$, divide the number line into three intervals: $(-\infty, -2)$, $(-2, 1)$, and $(1, \infty)$.

Interval Testing for y

Check the sign of $\frac{y - 1}{y+2}$ in each interval:

• Interval 1: $y < -2$. Pick $y=-3$: $\frac{-3 - 1}{-3 + 2} = \frac{-4}{-1} = 4$. $4$ is not less than $0$.
• Interval 2: $-2 < y < 1$. Pick $y=0$: $\frac{0 - 1}{0 + 2} = \frac{-1}{2}$. $-\frac{1}{2}$ is less than $0$. This interval satisfies the inequality.
• Interval 3: $y > 1$. Pick $y=2$: $\frac{2 - 1}{2 + 2} = \frac{1}{4}$. $\frac{1}{4}$ is not less than $0$.

Correct y Range Found

The interval satisfying $\frac{y - 1}{y+2} < 0$ is $-2 < y < 1$.

Therefore, the correct range for $y$ is:

$ -2 < y < 1 $