Sure, let's solve the inequality step-by-step:
Given inequality:
[tex]\[
\frac{x}{3} - 4 < -2
\][/tex]
Step 1: Isolate [tex]\(\frac{x}{3}\)[/tex].
First, add 4 to both sides of the inequality to move the constant term on the left side to the right side:
[tex]\[
\frac{x}{3} - 4 + 4 < -2 + 4
\][/tex]
[tex]\[
\frac{x}{3} < 2
\][/tex]
Step 2: Solve for [tex]\(x\)[/tex].
Next, multiply both sides of the inequality by 3 to eliminate the denominator:
[tex]\[
\left( \frac{x}{3} \right) \cdot 3 < 2 \cdot 3
\][/tex]
[tex]\[
x < 6
\][/tex]
So, the solution to the inequality is:
[tex]\[
x < 6
\][/tex]
Step 3: Graph the solution.
To graph the solution [tex]\(x < 6\)[/tex]:
- Draw a number line.
- Locate the point 6 on the number line.
- Since [tex]\(x\)[/tex] is less than 6, draw an open circle at 6 to indicate that 6 is not included in the solution.
- Shade the number line to the left of 6 to represent all values less than 6.
The graph looks like this:
[tex]\[
\begin{array}{cccccccccccc}
& & & & & & \circ & & & & & \\
\cdots & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & \cdots \\
\end{array}
\][/tex]
The shaded section to the left of 6 represents that all numbers less than 6 are solutions to the inequality.
