To solve the inequality [tex]\(-4x + 10 \geq 5x + 55\)[/tex], we will follow these steps:
1. Combine Like Terms:
We start by getting all the [tex]\(x\)[/tex]-terms on one side of the inequality and the constant terms on the other side.
[tex]\[
-4x + 10 \geq 5x + 55
\][/tex]
Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[
-4x - 5x + 10 \geq 55
\][/tex]
Simplify the left side:
[tex]\[
-9x + 10 \geq 55
\][/tex]
2. Isolate the Variable [tex]\(x\)[/tex]:
Next, we need to isolate [tex]\(x\)[/tex] by moving the constant term on the left side to the right side.
Subtract 10 from both sides:
[tex]\[
-9x + 10 - 10 \geq 55 - 10
\][/tex]
Simplify:
[tex]\[
-9x \geq 45
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Now, we divide both sides of the inequality by [tex]\(-9\)[/tex]. Remember, when we divide by a negative number, we must reverse the inequality sign.
[tex]\[
x \leq \frac{45}{-9}
\][/tex]
Simplify the fraction:
[tex]\[
x \leq -5
\][/tex]
4. Solution:
The solution to the inequality is [tex]\(x \leq -5\)[/tex].
5. Graph the Solution:
To graph this solution on the number line, we will:
- Draw a number line.
- Mark the point [tex]\(x = -5\)[/tex] on the number line with a closed circle (since [tex]\(\leq\)[/tex] includes the number [tex]\(-5\)[/tex]).
- Shade the number line to the left of [tex]\(-5\)[/tex] to indicate all values less than or equal to [tex]\(-5\)[/tex].
Here is a rough sketch of the number line:
```
<----|----|----|----|----|---->
-7 -6 -5 -4 -3 -2
●===========>
(Shaded region representing x ≤ -5)
```
Therefore, the solution to the inequality [tex]\(-4x + 10 \geq 5x + 55\)[/tex] is [tex]\(x \leq -5\)[/tex], and it can be represented on the number line as shown above.
