Solve Inequality: (3x-2)/(x+5) <= 2

Hey guys! Today we're diving deep into the world of algebra to tackle a juicy inequality problem. We're going to solve $\frac{3 x-2}{x+5} \leq 2$ step-by-step, making sure we understand every single move we make. This isn't just about getting the answer; it's about understanding the why behind it, which is super important for nailing those tough math problems. We'll break it down so clearly, you'll be solving inequalities like a pro in no time. So, grab your notebooks, get comfy, and let's unravel this algebraic mystery together! Remember, practice makes perfect, and by working through this, you're already ahead of the game.

Understanding the Inequality

Alright, let's get started by understanding the inequality we're working with: $\frac{3 x-2}{x+5} \leq 2$. The big challenge here is that we have a variable, 'x', in the denominator. This isn't like solving a simple equation where you can just multiply both sides by the denominator without a second thought. Why? Because the sign of the denominator can change depending on the value of 'x', and when you multiply or divide an inequality by a negative number, you have to flip the inequality sign. That's a major rule we cannot forget! If we were to just multiply both sides by $(x+5)$, we'd be assuming $(x+5)$ is positive, which isn't always true. So, we need a more robust method that handles all cases. The standard approach for these types of inequalities is to get everything on one side, find a common denominator, and then analyze the sign of the resulting expression. This method ensures we cover all possible scenarios for 'x' and arrive at the correct solution set. It’s all about being methodical and avoiding common pitfalls. We want to transform the inequality into a form where we can easily determine where the expression is less than or equal to zero. This typically involves moving the constant term to the left side and combining the terms into a single rational expression.

Step 1: Get Zero on One Side

The first crucial step in solving this type of inequality is to get a zero on one side. This is a standard technique that simplifies the analysis. So, we'll subtract 2 from both sides of the inequality:

$\frac{3 x-2}{x+5} - 2 \leq 0$

Now, we need to combine the terms on the left side into a single fraction. To do this, we'll find a common denominator, which is $(x+5)$. We rewrite 2 as $\frac{2(x+5)}{x+5}$:

$\frac{3 x-2}{x+5} - \frac{2(x+5)}{x+5} \leq 0$

Combine the numerators over the common denominator:

$\frac{(3 x-2) - 2(x+5)}{x+5} \leq 0$

Now, let's simplify the numerator. Distribute the -2:

$\frac{3 x-2 - 2x - 10}{x+5} \leq 0$

Combine like terms in the numerator:

$\frac{(3x - 2x) + (-2 - 10)}{x+5} \leq 0$

$\frac{x - 12}{x+5} \leq 0$

So, our original inequality $\frac{3 x-2}{x+5} \leq 2$ has been transformed into $\frac{x - 12}{x+5} \leq 0$. This form is much easier to work with because we're looking for where a rational expression is less than or equal to zero. This means we need to find the values of 'x' for which the numerator and denominator have opposite signs, or where the numerator is zero (but the denominator is not).

Step 2: Find Critical Points

Next up, guys, we need to find our critical points. These are the values of 'x' that make the numerator or the denominator equal to zero. These points are critical because they are the only places where the expression $\frac{x - 12}{x+5}$ can change its sign (from positive to negative or vice versa).

First, set the numerator equal to zero:

$x - 12 = 0$

Solving for 'x', we get:

$x = 12$

This is one of our critical points.

Second, set the denominator equal to zero:

$x + 5 = 0$

Solving for 'x', we get:

$x = -5$

This is our second critical point. Now, a really important thing to remember here: the value of 'x' that makes the denominator zero ($x = -5$ in this case) can never be part of the solution. Why? Because division by zero is undefined! So, $x = -5$ will always be an open circle on our number line, meaning it's excluded from the solution set. The value of 'x' that makes the numerator zero ($x = 12$) can be part of the solution if the inequality is "less than or equal to" or "greater than or equal to" (like ours is, $\leq$). So, $x = 12$ will be a closed circle on our number line, meaning it's included.

Step 3: Analyze the Sign of the Expression

We've got our critical points, $x = -5$ and $x = 12$. These points divide the number line into three distinct intervals:

1. $x < -5$
2. $-5 < x < 12$
3. $x > 12$

Our goal is to determine the sign of the expression $\frac{x - 12}{x+5}$ in each of these intervals. We want to find where this expression is less than or equal to zero.

We can use a test value from each interval to check the sign. Let's pick a value for 'x' that's less than -5, say $x = -6$:

Numerator: $(-6) - 12 = -18$ (Negative)

Denominator: $(-6) + 5 = -1$ (Negative)

Expression: $\frac{-18}{-1} = 18$ (Positive)

So, for $x < -5$, the expression is positive. This interval is not part of our solution.

Now, let's pick a test value between -5 and 12, say $x = 0$:

Numerator: $(0) - 12 = -12$ (Negative)

Denominator: $(0) + 5 = 5$ (Positive)

Expression: $\frac{-12}{5}$ (Negative)

So, for $-5 < x < 12$, the expression is negative. This interval is part of our solution because we want where the expression is $\leq 0$.

Finally, let's pick a test value greater than 12, say $x = 13$:

Numerator: $(13) - 12 = 1$ (Positive)

Denominator: $(13) + 5 = 18$ (Positive)

Expression: $\frac{1}{18}$ (Positive)

So, for $x > 12$, the expression is positive. This interval is not part of our solution.

This sign analysis is super important, guys. It's how we visually map out where our inequality holds true. We're essentially checking the