To solve the inequality [tex]\(-4x + 10 \geq 5x + 55\)[/tex], follow these steps:
1. Move the 'x' terms to one side:
Start by isolating the terms that contain [tex]\(x\)[/tex]. Add [tex]\(4x\)[/tex] to both sides of the inequality:
[tex]\[
-4x + 10 + 4x \geq 5x + 55 + 4x
\][/tex]
Simplifying this gives:
[tex]\[
10 \geq 9x + 55
\][/tex]
2. Move the constant terms to the other side:
Next, isolate the [tex]\(x\)[/tex] term by subtracting 55 from both sides:
[tex]\[
10 - 55 \geq 9x
\][/tex]
Simplifying this gives:
[tex]\[
-45 \geq 9x
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Divide both sides by 9 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{-45}{9} \geq x
\][/tex]
Simplifying this gives:
[tex]\[
-5 \geq x
\][/tex]
This is the same as:
[tex]\[
x \leq -5
\][/tex]
4. Graph the solution on the number line:
To graph [tex]\(x \leq -5\)[/tex] on the number line:
- Draw a number line.
- Make a point at [tex]\(-5\)[/tex].
- Shade the number line to the left of [tex]\(-5\)[/tex], including the point at [tex]\(-5\)[/tex] itself (which can be denoted by a closed dot to indicate that [tex]\(-5\)[/tex] is included in the solution).
Example:
```
<====●========================================>
-5
```
This indicates that all numbers less than or equal to [tex]\(-5\)[/tex] satisfy the inequality [tex]\(-4x + 10 \geq 5x + 55\)[/tex].
