Let's solve the inequality step-by-step and then graph its solution on the number line.
Given inequality:
[tex]\[ 3x - 11 > 7x + 9 \][/tex]
Step 1: Subtract [tex]\( 3x \)[/tex] from both sides to isolate the variable term on one side.
[tex]\[ -11 > 4x + 9 \][/tex]
Step 2: Subtract [tex]\( 9 \)[/tex] from both sides to isolate the constant term.
[tex]\[ -20 > 4x \][/tex]
Step 3: Divide both sides by [tex]\( 4 \)[/tex] to solve for [tex]\( x \)[/tex].
[tex]\[ -5 > x \][/tex]
or equivalently,
[tex]\[ x < -5 \][/tex]
The solution [tex]\( x < -5 \)[/tex] means that [tex]\( x \)[/tex] is any number less than [tex]\(-5\)[/tex].
To graph this inequality on a number line:
1. Draw a number line.
2. Identify the point [tex]\( -5 \)[/tex] on the number line.
3. Place an open circle at [tex]\( -5 \)[/tex] to indicate that [tex]\( -5 \)[/tex] is not included in the solution set.
4. Shade everything to the left of [tex]\( -5 \)[/tex] to show that the solution includes all values less than [tex]\( -5 \)[/tex].
Here is the graphical representation of the solution:
[tex]\[
\begin{array}{c|ccccccccccccccccc}
\text{Number line:} & \cdots & -8 & -7 & -6 & -5 & -4 & -3 & \cdots \\
\hline
\text{Graph:} & & \bullet & \bullet & \bullet & \circ & & &
\end{array}
\rightarrow
\begin{array}{c}
\text{\ \ \ \ \ \ \ \ \ \ \ \ \ Ray left of -5}
\end{array}
\][/tex]
To detail:
- "•" represents the points to the left of [tex]\(-5\)[/tex] being part of the solution.
- "o" represents the open circle at [tex]\(-5\)[/tex], showing it is not included.
This accurately represents the solution set to the inequality [tex]\( 3x - 11 > 7x + 9 \)[/tex].
