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Sagot :
Certainly! Let's break down the steps to solve the inequality [tex]\(\frac{3}{x+2} + 4 \geq 3\)[/tex] and explain why the solution does not include [tex]\(x = -2\)[/tex].
### Step-by-Step Solution:
1. Subtract 4 from both sides:
[tex]\[\frac{3}{x+2} + 4 - 4 \geq 3 - 4\][/tex]
Simplifying this expression, we get:
[tex]\[\frac{3}{x+2} \geq -1\][/tex]
2. Isolate the term involving [tex]\(x\)[/tex]:
To do this, let's consider the inequality:
[tex]\[ \frac{3}{x+2} \geq -1 \][/tex]
We need to remove the denominator on the left side. We'll multiply both sides of the inequality by [tex]\(x+2\)[/tex], but we must be careful: the sign of [tex]\(x+2\)[/tex] could be positive or negative, which affects our inequality direction.
[tex]\[ \begin{cases} \frac{3}{x+2} \cdot (x+2) \geq -1 \cdot (x+2) & \text{if } x \neq -2 \end{cases} \][/tex]
3. Simplify and solve for [tex]\(x\)[/tex]:
The term [tex]\(\frac{3}{x+2} \cdot (x+2)\)[/tex] simplifies to 3, providing:
[tex]\[ 3 \geq -1(x + 2) \][/tex]
Distribute [tex]\(-1\)[/tex] on the right side:
[tex]\[ 3 \geq -x - 2 \][/tex]
Add 2 to both sides to simplify:
[tex]\[ 3 + 2 \geq -x \][/tex]
[tex]\[ 5 \geq -x \][/tex]
Multiply both sides by [tex]\(-1\)[/tex] (remember to reverse the inequality sign):
[tex]\[ -5 \leq x \][/tex]
Or,
[tex]\[ x \geq -5 \][/tex]
4. Consider [tex]\(x = -2\)[/tex]:
Throughout these steps, an important factor is that [tex]\(x \neq -2\)[/tex] because if [tex]\(x = -2\)[/tex], the denominator in the original inequality [tex]\(\frac{3}{x+2} + 4 \geq 3\)[/tex] would become zero, which would make the fraction undefined.
[tex]\[ \frac{3}{x + 2} \quad \text{is undefined when } x = -2 \][/tex]
### Conclusion:
The solution to the inequality [tex]\(\frac{3}{x+2} + 4 \geq 3\)[/tex] is [tex]\(x \geq -5\)[/tex], but not including [tex]\(x = -2\)[/tex], because at [tex]\(x = -2\)[/tex] the term [tex]\(\frac{3}{x+2}\)[/tex] is undefined (it involves division by zero).
Thus, the range of [tex]\(x\)[/tex] is:
[tex]\[ x \in [-5, \infty) \quad \text{excluding} \quad x = -2 \][/tex]
This exclusion of [tex]\(x = -2\)[/tex] is why there is an open circle at [tex]\(x = -2\)[/tex] on the graph of the solution. An open circle represents a value that is not included in the solution set.
### Step-by-Step Solution:
1. Subtract 4 from both sides:
[tex]\[\frac{3}{x+2} + 4 - 4 \geq 3 - 4\][/tex]
Simplifying this expression, we get:
[tex]\[\frac{3}{x+2} \geq -1\][/tex]
2. Isolate the term involving [tex]\(x\)[/tex]:
To do this, let's consider the inequality:
[tex]\[ \frac{3}{x+2} \geq -1 \][/tex]
We need to remove the denominator on the left side. We'll multiply both sides of the inequality by [tex]\(x+2\)[/tex], but we must be careful: the sign of [tex]\(x+2\)[/tex] could be positive or negative, which affects our inequality direction.
[tex]\[ \begin{cases} \frac{3}{x+2} \cdot (x+2) \geq -1 \cdot (x+2) & \text{if } x \neq -2 \end{cases} \][/tex]
3. Simplify and solve for [tex]\(x\)[/tex]:
The term [tex]\(\frac{3}{x+2} \cdot (x+2)\)[/tex] simplifies to 3, providing:
[tex]\[ 3 \geq -1(x + 2) \][/tex]
Distribute [tex]\(-1\)[/tex] on the right side:
[tex]\[ 3 \geq -x - 2 \][/tex]
Add 2 to both sides to simplify:
[tex]\[ 3 + 2 \geq -x \][/tex]
[tex]\[ 5 \geq -x \][/tex]
Multiply both sides by [tex]\(-1\)[/tex] (remember to reverse the inequality sign):
[tex]\[ -5 \leq x \][/tex]
Or,
[tex]\[ x \geq -5 \][/tex]
4. Consider [tex]\(x = -2\)[/tex]:
Throughout these steps, an important factor is that [tex]\(x \neq -2\)[/tex] because if [tex]\(x = -2\)[/tex], the denominator in the original inequality [tex]\(\frac{3}{x+2} + 4 \geq 3\)[/tex] would become zero, which would make the fraction undefined.
[tex]\[ \frac{3}{x + 2} \quad \text{is undefined when } x = -2 \][/tex]
### Conclusion:
The solution to the inequality [tex]\(\frac{3}{x+2} + 4 \geq 3\)[/tex] is [tex]\(x \geq -5\)[/tex], but not including [tex]\(x = -2\)[/tex], because at [tex]\(x = -2\)[/tex] the term [tex]\(\frac{3}{x+2}\)[/tex] is undefined (it involves division by zero).
Thus, the range of [tex]\(x\)[/tex] is:
[tex]\[ x \in [-5, \infty) \quad \text{excluding} \quad x = -2 \][/tex]
This exclusion of [tex]\(x = -2\)[/tex] is why there is an open circle at [tex]\(x = -2\)[/tex] on the graph of the solution. An open circle represents a value that is not included in the solution set.
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