To solve the given inequalities, let's take it step by step and ensure we understand the logic behind the solution.
### Step 1: Given Inequalities
We start with the given inequalities:
[tex]\[
-3x + 7 > -1 \text{ or } -3x + 7 < 1
\][/tex]
### Step 2: Solving Each Inequality Separately
Let's solve each part of the inequality separately:
#### For [tex]\( -3x + 7 > -1 \)[/tex]:
1. Subtract 7 from both sides:
[tex]\[
-3x + 7 - 7 > -1 - 7
\][/tex]
[tex]\[
-3x > -8
\][/tex]
2. Divide both sides by -3 (and remember to reverse the inequality sign because we are dividing by a negative number):
[tex]\[
x < \frac{8}{3}
\][/tex]
#### For [tex]\( -3x + 7 < 1 \)[/tex]:
1. Subtract 7 from both sides:
[tex]\[
-3x + 7 - 7 < 1 - 7
\][/tex]
[tex]\[
-3x < -6
\][/tex]
2. Divide both sides by -3 (again, reversing the inequality sign):
[tex]\[
x > 2
\][/tex]
### Step 3: Combining the Solutions
From the two parts, you get:
[tex]\[
x < \frac{8}{3} \text{ or } x > 2
\][/tex]
### Step 4: Interval Notation
To write this solution in interval notation, we need to describe it using the appropriate intervals:
[tex]\[
(-\infty, \frac{8}{3}) \cup (2, \infty)
\][/tex]
### Step 5: Graphing the Solution
To graph the solution, you can draw a number line and shade the regions as follows:
1. Shade the region to the left of [tex]\( \frac{8}{3} \)[/tex]. This includes all values less than [tex]\( \frac{8}{3} \)[/tex], but not [tex]\( \frac{8}{3} \)[/tex] itself.
2. Shade the region to the right of [tex]\( 2 \)[/tex]. This includes all values greater than 2, but not 2 itself.
There will be a gap between [tex]\( 2 \)[/tex] and [tex]\( \frac{8}{3} \)[/tex] because these two intervals do not overlap.
### Final Answer
Therefore, the solution set for the inequality [tex]\( -3x + 7 > -1 \)[/tex] or [tex]\( -3x + 7 < 1 \)[/tex] is:
[tex]\[
(-\infty, \frac{8}{3}) \cup (2, \infty)
\][/tex]
This can be interpreted as any [tex]\( x \)[/tex] that is either less than [tex]\( \frac{8}{3} \)[/tex] or greater than 2.
