Question
x^2+8x+16>=0 graph each inequality
Answers
GPT-4o mini
To solve the inequality \( x^2 + 8x + 16 \geq 0 \), we can start by recognizing that the left-hand side is a quadratic expression. We can factor it:
\[
x^2 + 8x + 16 = (x + 4)^2
\]
Now, we rewrite the inequality:
\[
(x + 4)^2 \geq 0
\]
The expression \((x + 4)^2\) represents a perfect square, which is always greater than or equal to zero for all real numbers \(x\). The only time it equals zero is when \(x + 4 = 0\), which occurs at:
\[
x = -4
\]
### Step 1: Analyze the solution
- For any \(x\) other than \(-4\), \((x + 4)^2 > 0\).
- At \(x = -4\), \((x + 4)^2 = 0\).
### Step 2: Write the solution
Thus, the solution to the inequality \(x^2 + 8x + 16 \geq 0\) is:
\[
x \in (-\infty, -4] \cup (-4, \infty)
\]
### Step 3: Graph the inequality
1. **Draw the x-axis and y-axis**.
2. **Plot the vertex** of the parabola, which is at the point \((-4, 0)\), where the curve touches the x-axis.
3. **Draw the parabola** that opens upwards since the coefficient of \(x^2\) is positive. The curve will touch the x-axis at \((-4, 0)\) and open upward on both sides.
4. **Shade the area** above the x-axis to indicate where \( (x + 4)^2 \) is greater than 0 and include the point \((-4, 0)\) to show that it is part of the solution set.
The final graph will indicate that the inequality \(x^2 + 8x + 16 \geq 0\) holds true for all \(x\), with the equality point at \(x = -4\).
\[
x^2 + 8x + 16 = (x + 4)^2
\]
Now, we rewrite the inequality:
\[
(x + 4)^2 \geq 0
\]
The expression \((x + 4)^2\) represents a perfect square, which is always greater than or equal to zero for all real numbers \(x\). The only time it equals zero is when \(x + 4 = 0\), which occurs at:
\[
x = -4
\]
### Step 1: Analyze the solution
- For any \(x\) other than \(-4\), \((x + 4)^2 > 0\).
- At \(x = -4\), \((x + 4)^2 = 0\).
### Step 2: Write the solution
Thus, the solution to the inequality \(x^2 + 8x + 16 \geq 0\) is:
\[
x \in (-\infty, -4] \cup (-4, \infty)
\]
### Step 3: Graph the inequality
1. **Draw the x-axis and y-axis**.
2. **Plot the vertex** of the parabola, which is at the point \((-4, 0)\), where the curve touches the x-axis.
3. **Draw the parabola** that opens upwards since the coefficient of \(x^2\) is positive. The curve will touch the x-axis at \((-4, 0)\) and open upward on both sides.
4. **Shade the area** above the x-axis to indicate where \( (x + 4)^2 \) is greater than 0 and include the point \((-4, 0)\) to show that it is part of the solution set.
The final graph will indicate that the inequality \(x^2 + 8x + 16 \geq 0\) holds true for all \(x\), with the equality point at \(x = -4\).