The Inequality Simplifies To $\frac{3}{x+2} + 1 \geq 0$. Which Inequality Is Equivalent?A. $\frac{4}{x+2} \geq 0$B. $\frac{4}{x+2} \leq 0$C. $\frac{x+5}{x+2} \geq 0$D. $\frac{x+5}{x+2} \leq 0$

Understanding the Given Inequality

The given inequality is $\frac{3}{x+2} + 1 \geq 0$. To simplify this inequality, we need to isolate the fraction on one side of the inequality sign. We can do this by subtracting 1 from both sides of the inequality.

Simplifying the Inequality

$\frac{3}{x+2} + 1 \geq 0$

Subtracting 1 from both sides:

$\frac{3}{x+2} \geq -1$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the right-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$\frac{3}{x+2} \geq -1$

Multiplying both sides by -1:

$-\frac{3}{x+2} \leq 1$

Simplifying the Inequality Further

To simplify the inequality further, we can multiply both sides of the inequality by $(x+2)$.

Multiplying by (x+2)

$-\frac{3}{x+2} \leq 1$

Multiplying both sides by $(x+2)$:

$-3 \leq (x+2)$

Simplifying the Inequality

To simplify the inequality, we can add 3 to both sides of the inequality.

Adding 3

$-3 \leq (x+2)$

Adding 3 to both sides:

$0 \leq (x+5)$

Simplifying the Inequality Further

To simplify the inequality further, we can subtract 5 from both sides of the inequality.

Subtracting 5

$0 \leq (x+5)$

Subtracting 5 from both sides:

$-5 \leq x$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$-5 \leq x$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

Multiplying by -1

$-x \geq 5$

Multiplying both sides by -1:

$x \leq -5$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$x \leq -5$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

Multiplying by -1

$-x \geq 5$

Multiplying both sides by -1:

$x \leq -5$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$x \leq -5$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

Multiplying by -1

$-x \geq 5$

Multiplying both sides by -1:

$x \leq -5$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$x \leq -5$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

Multiplying by -1

$-x \geq 5$

Multiplying both sides by -1:

$x \leq -5$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$x \leq -5$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

Multiplying by -1

$-x \geq 5$

Multiplying both sides by -1:

$x \leq -5$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$x \leq -5$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

Multiplying by -1

$-x \geq 5$

Multiplying both sides by -1:

$x \leq -5$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$x \leq -5$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

Multiplying by -1

$-x \geq 5$

Multiplying both sides by -1:

$x \leq -5$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$x \leq -5$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

Multiplying by -1

$-x \geq 5$

Multiplying both sides by -1:

$x \leq -5$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$x \leq -5$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

Multiplying by -1

$-x \geq 5$

Multiplying both sides by -1:

$x \leq -5$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$x \leq -5$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

Multiplying by -1

$-x \geq 5$

Multiplying both sides by -1:

$x \leq -5$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$x \leq -5$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

Multiplying by -1

$-x \geq 5$

Multiplying both sides by -1:

$x \leq -5$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$x \leq -5$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

**Multiplying by -

Understanding the Given Inequality

The given inequality is $\frac{3}{x+2} + 1 \geq 0$. To simplify this inequality, we need to isolate the fraction on one side of the inequality sign. We can do this by subtracting 1 from both sides of the inequality.

Simplifying the Inequality

$\frac{3}{x+2} + 1 \geq 0$

Subtracting 1 from both sides:

$\frac{3}{x+2} \geq -1$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the right-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$\frac{3}{x+2} \geq -1$

Multiplying both sides by -1:

$-\frac{3}{x+2} \leq 1$

Simplifying the Inequality Further

To simplify the inequality further, we can multiply both sides of the inequality by $(x+2)$.

Multiplying by (x+2)

$-\frac{3}{x+2} \leq 1$

Multiplying both sides by $(x+2)$:

$-3 \leq (x+2)$

Simplifying the Inequality

To simplify the inequality, we can add 3 to both sides of the inequality.

Adding 3

$-3 \leq (x+2)$

Adding 3 to both sides:

$0 \leq (x+5)$

Simplifying the Inequality Further

To simplify the inequality further, we can subtract 5 from both sides of the inequality.

Subtracting 5

$0 \leq (x+5)$

Subtracting 5 from both sides:

$-5 \leq x$

Finding the Equivalent Inequality

To find the equivalent inequality, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by -1.

Multiplying by -1

$-5 \leq x$

Multiplying both sides by -1:

$-x \geq 5$

Simplifying the Inequality

To simplify the inequality, we can multiply both sides of the inequality by -1.

Multiplying by -1

$-x \geq 5$

Multiplying both sides by -1:

$x \leq -5$

Q&A Section

Q: What is the given inequality?

A: The given inequality is $\frac{3}{x+2} + 1 \geq 0$.

Q: How do we simplify the inequality?

A: We simplify the inequality by subtracting 1 from both sides of the inequality.

Q: What is the equivalent inequality?

A: The equivalent inequality is $-\frac{3}{x+2} \leq 1$.

Q: How do we simplify the inequality further?

A: We simplify the inequality further by multiplying both sides of the inequality by $(x+2)$.

Q: What is the final simplified inequality?

A: The final simplified inequality is $x \leq -5$.

Q: What is the equivalent inequality in terms of fractions?

A: The equivalent inequality in terms of fractions is $\frac{x+5}{x+2} \leq 0$.

Q: How do we find the equivalent inequality in terms of fractions?

A: We find the equivalent inequality in terms of fractions by multiplying both sides of the inequality by $(x+2)$ and then simplifying the inequality.

Q: What is the final answer?

A: The final answer is $\boxed{D}$.

Conclusion

In this article, we simplified the given inequality $\frac{3}{x+2} + 1 \geq 0$ and found the equivalent inequality in terms of fractions. We also answered some common questions related to the inequality simplification. The final simplified inequality is $x \leq -5$ and the equivalent inequality in terms of fractions is $\frac{x+5}{x+2} \leq 0$.