Sure, let's solve the inequality step-by-step and represent the solution on a number line.
1. Original Inequality:
[tex]\[
3x - 11 > 7x + 9
\][/tex]
2. Isolate the variable [tex]\(x\)[/tex]:
- Subtract [tex]\(7x\)[/tex] from both sides:
[tex]\[
3x - 11 - 7x > 9
\][/tex]
Simplify:
[tex]\[
-4x - 11 > 9
\][/tex]
- Add 11 to both sides:
[tex]\[
-4x - 11 + 11 > 9 + 11
\][/tex]
Simplify:
[tex]\[
-4x > 20
\][/tex]
- Divide both sides by -4. Remember, dividing by a negative number reverses the inequality:
[tex]\[
x < -5
\][/tex]
Thus, our solution set is [tex]\(x < -5\)[/tex].
3. Represent the Solution on a Number Line:
- We need to draw a number line and shade the region representing [tex]\(x < -5\)[/tex].
- Mark -5 on the number line.
- Since the inequality is strict (does not include -5), we use an open circle at -5.
- Shade the region to the left of -5 (indicating all numbers less than -5).
Here is the graphical representation on the number line:
[tex]\[
\begin{array}{c}
\text{Number Line:} \quad \\
\overset { \xrightarrow {\hspace{2mm}} \xleftarrow{ \hspace{2mm}}}{- \infty \quad \quad \begin{array}{l}
\begin{array}{|c|}
\hline
-5\\
\hline
\end{array} \ \ \ \\
\end{array} \quad \quad \quad \infty} \\
\end{array}
\][/tex]
Explanation:
- The open circle at -5 indicates that -5 is not included in the solution set.
- The shading to the left indicates that all values less than -5 satisfy the inequality [tex]\(3x - 11 > 7x + 9\)[/tex].
Thus, the solution to the inequality is all real numbers [tex]\(x\)[/tex] such that [tex]\(x < -5\)[/tex].
