Sure! Let's solve the inequality step by step and then graph the solution on a number line.
The inequality we need to solve is:
[tex]\[ -2x + 8 > 3x + 10 \][/tex]
### Step 1: Simplify the Inequality
First, we need to get all the terms involving [tex]\( x \)[/tex] on one side and the constant terms on the other side of the inequality.
To do this, we can add [tex]\( 2x \)[/tex] to both sides of the inequality to eliminate the [tex]\( -2x \)[/tex] term on the left-hand side:
[tex]\[ -2x + 8 + 2x > 3x + 10 + 2x \][/tex]
[tex]\[ 8 > 5x + 10 \][/tex]
Now, we need to isolate [tex]\( x \)[/tex] on one side. We can start by subtracting 10 from both sides:
[tex]\[ 8 - 10 > 5x + 10 - 10 \][/tex]
[tex]\[ -2 > 5x \][/tex]
### Step 2: Solve for [tex]\( x \)[/tex]
Now, we need to isolate [tex]\( x \)[/tex]. To do this, we divide both sides by 5:
[tex]\[ \frac{-2}{5} > x \][/tex]
### Step 3: Rewrite the Solution
The inequality can be rewritten as:
[tex]\[ x < \frac{-2}{5} \][/tex]
This is the solution to the inequality:
[tex]\[ x < -0.4 \][/tex]
### Step 4: Graph the Solution
To graph this solution on a number line:
1. We draw a number line.
2. Identify the point [tex]\( -0.4 \)[/tex] on the number line.
3. Since the inequality is [tex]\( x < -0.4 \)[/tex], we will indicate that all values to the left of [tex]\( -0.4 \)[/tex] are solutions. To do this, we use an open circle at [tex]\( -0.4 \)[/tex] (indicating that [tex]\( -0.4 \)[/tex] itself is not included in the solutions) and shade the line to the left of [tex]\( -0.4 \)[/tex].
Here is a visual representation:
[tex]\[ \dots\rightarrow \circ \quad -0.4 \quad--------- \][/tex]
In summary, the solution to the inequality [tex]\( -2x + 8 > 3x + 10 \)[/tex] is [tex]\( x < -0.4 \)[/tex], and it is graphically represented as an open circle at [tex]\( -0.4 \)[/tex] with shading to the left to indicate all numbers less than [tex]\( -0.4 \)[/tex].
