Question

Solve the following inequality and graph the solution set on the number line.
2(x-3)-5x ≤9
     
-10-9-8-7-6-5-4-3-2-1012345678910

Answer 1

To solve the inequality \(2(x-3) - 5x \leq 9\), let's start by simplifying the left side.

1. Distribute the \(2\):
\[
2(x - 3) = 2x - 6
\]
Therefore, the inequality becomes:
\[
2x - 6 - 5x \leq 9
\]

2. Combine like terms:
\[
-3x - 6 \leq 9
\]

3. Add \(6\) to both sides:
\[
-3x \leq 15
\]

4. Divide both sides by \(-3\). Remember, when we divide or multiply by a negative number, we reverse the inequality sign:
\[
x \geq -5
\]

Now, the solution set for the inequality is \(x \geq -5\).

Next, let's graph the solution on the number line. We will indicate that \(x\) includes \(-5\) and everything to the right (larger values).

Here's the graphical representation:

```
<----|----|----|----|----|----|----|----|----|----|----|---->
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
●-------------------->
```

The dot at \(-5\) indicates that it is included in the solution (closed dot), and the arrow to the right signifies that all numbers greater than \(-5\) are also part of the solution set.