Let's solve the inequality step-by-step:
### Step 1: Start with the given inequality
[tex]\[
-2x + 4 > -6
\][/tex]
### Step 2: Isolate the term with [tex]\( x \)[/tex]
First, we need to isolate the term containing the variable [tex]\( x \)[/tex]. To do this, subtract 4 from both sides of the inequality.
[tex]\[
-2x + 4 - 4 > -6 - 4
\][/tex]
Simplify both sides:
[tex]\[
-2x > -10
\][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Next, we need to solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex] on one side of the inequality. To do this, divide both sides by -2. It's important to remember that dividing both sides of an inequality by a negative number reverses the direction of the inequality.
[tex]\[
\frac{-2x}{-2} < \frac{-10}{-2}
\][/tex]
Simplify:
[tex]\[
x < 5
\][/tex]
So, the solution to the inequality is:
[tex]\[
x < 5
\][/tex]
### Step 4: Graph the solution
To graph the solution [tex]\( x < 5 \)[/tex] on a number line:
1. Draw a number line with relevant points marked.
2. Identify the point [tex]\( 5 \)[/tex] on the number line.
3. Since [tex]\( x \)[/tex] is less than 5 and does not include 5, use an open circle at [tex]\( 5 \)[/tex].
4. Shade the number line to the left of [tex]\( 5 \)[/tex] to indicate all numbers less than 5 are included in the solution.
Here's a rough sketch of the graph:
[tex]\[
\quad \quad \quad \quad \circ \quad \quad \quad \quad \quad \quad \quad \quad \quad
\quad \quad <--- \text{shade this region} \quad \quad < ----------------------> \quad \quad
\quad \quad \quad \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad 2 \quad \quad \quad 3 \quad \quad 4 \quad (5) \quad 6 \quad
\][/tex]
In conclusion, the solution to the inequality [tex]\( -2x + 4 > -6 \)[/tex] is [tex]\( x < 5 \)[/tex]. The graph on a number line represents all values of [tex]\( x \)[/tex] that are less than 5.
