Certainly! Let's solve the given inequality step-by-step:
The inequality provided is:
[tex]\[ -10 < 3x - 4 < 8 \][/tex]
Step 1: Isolate the term involving the variable [tex]\( x \)[/tex] in the inequality.
First, add 4 to all parts of the inequality:
[tex]\[ -10 + 4 < 3x - 4 + 4 < 8 + 4 \][/tex]
[tex]\[ -6 < 3x < 12 \][/tex]
Step 2: Solve for [tex]\( x \)[/tex] by dividing all parts of the inequality by 3.
Divide each term by 3:
[tex]\[ \frac{-6}{3} < \frac{3x}{3} < \frac{12}{3} \][/tex]
[tex]\[ -2 < x < 4 \][/tex]
So, the solution to the inequality [tex]\( -10 < 3x - 4 < 8 \)[/tex] is:
[tex]\[ -2 < x < 4 \][/tex]
Step 3: Represent the solution set graphically.
This means [tex]\( x \)[/tex] can take any value between [tex]\(-2\)[/tex] and [tex]\(4\)[/tex], but not including [tex]\(-2\)[/tex] and [tex]\(4\)[/tex].
On a number line, this is represented by:
- An open circle at [tex]\(-2\)[/tex] indicating that [tex]\(-2\)[/tex] is not included.
- An open circle at [tex]\(4\)[/tex] indicating that [tex]\(4\)[/tex] is not included.
- A shaded line between [tex]\(-2\)[/tex] and [tex]\(4\)[/tex] indicating all the values of [tex]\( x \)[/tex] between these points are included in the solution set.
Therefore, the graph of the solution set of [tex]\(-10 < 3x - 4 < 8\)[/tex] is a line segment on the number line extending from [tex]\(-2\)[/tex] to [tex]\(4\)[/tex] with open circles at each endpoint.
