Question
Solve for x and graph the solution on the number line below. the number line goes from -12 all the way to 12, please include how I should color the number line, -21≥-7x
Answers
GPT-4o mini
To solve the inequality \(-21 \geq -7x\), we first want to isolate \(x\). We start by dividing both sides of the inequality by \(-7\). Remember that when we divide or multiply both sides of an inequality by a negative number, we must flip the inequality sign.
Here's how that looks:
\[
-21 \geq -7x \quad \text{(original inequality)}
\]
Dividing by \(-7\) and flipping the inequality sign gives:
\[
\frac{-21}{-7} \leq x
\]
This simplifies to:
\[
3 \leq x
\]
or equivalently:
\[
x \geq 3
\]
Now, to graph the solution on the number line:
1. Draw a number line that includes all integers from \(-12\) to \(12\).
2. Identify the point where \(x = 3\) on the number line.
3. Since \(x\) is greater than or equal to \(3\), we will make a **closed circle** at \(3\) to indicate that \(3\) is included in the solution set.
4. Color the portion of the line to the right of \(3\) (i.e., all values greater than \(3\)) to show that those values are part of the solution.
Here's the representation of your solution on the number line:
- A closed circle at \(3\).
- Shading to the right of \(3\):
```
<---|---|---|---|---|---|---|---|---|---|---|---|--->
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12
●───────────→
```
In this representation:
- The closed circle (●) at \(3\) indicates that \(3\) is included in the solution.
- The arrow (→) shows that all numbers greater than \(3\) are included in the solution set.
Here's how that looks:
\[
-21 \geq -7x \quad \text{(original inequality)}
\]
Dividing by \(-7\) and flipping the inequality sign gives:
\[
\frac{-21}{-7} \leq x
\]
This simplifies to:
\[
3 \leq x
\]
or equivalently:
\[
x \geq 3
\]
Now, to graph the solution on the number line:
1. Draw a number line that includes all integers from \(-12\) to \(12\).
2. Identify the point where \(x = 3\) on the number line.
3. Since \(x\) is greater than or equal to \(3\), we will make a **closed circle** at \(3\) to indicate that \(3\) is included in the solution set.
4. Color the portion of the line to the right of \(3\) (i.e., all values greater than \(3\)) to show that those values are part of the solution.
Here's the representation of your solution on the number line:
- A closed circle at \(3\).
- Shading to the right of \(3\):
```
<---|---|---|---|---|---|---|---|---|---|---|---|--->
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12
●───────────→
```
In this representation:
- The closed circle (●) at \(3\) indicates that \(3\) is included in the solution.
- The arrow (→) shows that all numbers greater than \(3\) are included in the solution set.
Inequality Notation is?
GPT-4o mini
The solution to the inequality \(-21 \geq -7x\) can be expressed in inequality notation as:
\[
x \geq 3
\]
This means that \(x\) can take any value that is equal to \(3\) or greater than \(3\). In interval notation, this would be represented as:
\[
[3, \infty)
\]
Here, the square bracket indicates that \(3\) is included in the set of solutions, while the parentheses indicate that the set continues indefinitely toward positive infinity.
\[
x \geq 3
\]
This means that \(x\) can take any value that is equal to \(3\) or greater than \(3\). In interval notation, this would be represented as:
\[
[3, \infty)
\]
Here, the square bracket indicates that \(3\) is included in the set of solutions, while the parentheses indicate that the set continues indefinitely toward positive infinity.