Question
What is x²-5x>0 supposed to look like on the number line
Answers
GPT-4o mini
To solve the inequality \( x^2 - 5x > 0 \), we first need to factor the left-hand side:
\[
x^2 - 5x = x(x - 5)
\]
We can rewrite the inequality as:
\[
x(x - 5) > 0
\]
Next, we find the critical points by setting the expression equal to zero:
\[
x(x - 5) = 0
\]
This gives us the solutions:
\[
x = 0 \quad \text{and} \quad x = 5
\]
These critical points divide the number line into three intervals:
1. \( (-\infty, 0) \)
2. \( (0, 5) \)
3. \( (5, \infty) \)
Now we will test a point from each interval to determine where the inequality \( x(x - 5) > 0 \) holds true.
1. **Interval \( (-\infty, 0) \)**:
- Choose \( x = -1 \):
\[
(-1)(-1 - 5) = (-1)(-6) = 6 > 0
\]
So, this interval satisfies the inequality.
2. **Interval \( (0, 5) \)**:
- Choose \( x = 1 \):
\[
(1)(1 - 5) = (1)(-4) = -4 < 0
\]
So, this interval does not satisfy the inequality.
3. **Interval \( (5, \infty) \)**:
- Choose \( x = 6 \):
\[
(6)(6 - 5) = (6)(1) = 6 > 0
\]
So, this interval satisfies the inequality.
Now we can summarize the results:
- The inequality \( x^2 - 5x > 0 \) is satisfied in the intervals \( (-\infty, 0) \) and \( (5, \infty) \).
On the number line, this can be represented as:
- Open interval from \( -\infty \) to \( 0 \) (not including \( 0 \))
- Open interval from \( 5 \) to \( \infty \) (not including \( 5 \))
To visualize this, you would have:
- An arrow or a line going to the left from \( 0 \) (not including \( 0 \))
- An arrow or a line going to the right from \( 5 \) (not including \( 5 \))
Putting it all together, the solution on the number line looks like this:
\[
\text{Number Line: } (-\infty \quad \text{---} \quad 0 \quad \text{---} \quad 5 \quad \text{---} \quad \infty)
\]
Where the areas before \( 0 \) and after \( 5 \) are shaded to indicate they are included in the solution.
\[
x^2 - 5x = x(x - 5)
\]
We can rewrite the inequality as:
\[
x(x - 5) > 0
\]
Next, we find the critical points by setting the expression equal to zero:
\[
x(x - 5) = 0
\]
This gives us the solutions:
\[
x = 0 \quad \text{and} \quad x = 5
\]
These critical points divide the number line into three intervals:
1. \( (-\infty, 0) \)
2. \( (0, 5) \)
3. \( (5, \infty) \)
Now we will test a point from each interval to determine where the inequality \( x(x - 5) > 0 \) holds true.
1. **Interval \( (-\infty, 0) \)**:
- Choose \( x = -1 \):
\[
(-1)(-1 - 5) = (-1)(-6) = 6 > 0
\]
So, this interval satisfies the inequality.
2. **Interval \( (0, 5) \)**:
- Choose \( x = 1 \):
\[
(1)(1 - 5) = (1)(-4) = -4 < 0
\]
So, this interval does not satisfy the inequality.
3. **Interval \( (5, \infty) \)**:
- Choose \( x = 6 \):
\[
(6)(6 - 5) = (6)(1) = 6 > 0
\]
So, this interval satisfies the inequality.
Now we can summarize the results:
- The inequality \( x^2 - 5x > 0 \) is satisfied in the intervals \( (-\infty, 0) \) and \( (5, \infty) \).
On the number line, this can be represented as:
- Open interval from \( -\infty \) to \( 0 \) (not including \( 0 \))
- Open interval from \( 5 \) to \( \infty \) (not including \( 5 \))
To visualize this, you would have:
- An arrow or a line going to the left from \( 0 \) (not including \( 0 \))
- An arrow or a line going to the right from \( 5 \) (not including \( 5 \))
Putting it all together, the solution on the number line looks like this:
\[
\text{Number Line: } (-\infty \quad \text{---} \quad 0 \quad \text{---} \quad 5 \quad \text{---} \quad \infty)
\]
Where the areas before \( 0 \) and after \( 5 \) are shaded to indicate they are included in the solution.