Sure, let's solve the inequality step-by-step and find the correct solution in interval notation.
Given the inequality:
[tex]\[ 9x - 8 \leq 4x - 15 \][/tex]
1. Move all terms involving [tex]\( x \)[/tex] to one side of the inequality:
[tex]\[ 9x - 4x - 8 \leq -15 \][/tex]
[tex]\[ 5x - 8 \leq -15 \][/tex]
2. Isolate the [tex]\( x \)[/tex]-term by adding 8 to both sides of the inequality:
[tex]\[ 5x - 8 + 8 \leq -15 + 8 \][/tex]
[tex]\[ 5x \leq -7 \][/tex]
3. Solve for [tex]\( x \)[/tex] by dividing both sides by 5:
[tex]\[ x \leq \frac{-7}{5} \][/tex]
[tex]\[ x \leq -1.4 \][/tex]
Thus, the solution set in interval notation is:
[tex]\[ (-\infty, -1.4] \][/tex]
Let’s graph the solution set on a number line:
1. Draw a number line and mark the point [tex]\(-1.4\)[/tex].
2. Use a filled (or solid) circle at [tex]\( -1.4 \)[/tex] to indicate that [tex]\(-1.4\)[/tex] is included in the solution set (i.e., [tex]\( x \)[/tex] can be equal to [tex]\(-1.4\)[/tex]).
3. Shade or draw a line extending leftwards from [tex]\(-1.4\)[/tex] to indicate that all values less than [tex]\(-1.4\)[/tex] are part of the solution set.
Here is the graphical representation:
[tex]\[
\begin{array}{c c c c c c c c c c c c c c c c c c c c c}
\textemdash & \textemdash & \textemdash & \textemdash & \bullet & \textemdash & \textemdash & \textemdash & \textemdash & \textemdash \\
& & & & -1.4 & & & & & \\
\end{array}
\][/tex]
Hence, the correct choice is:
A. The solution set in interval notation is [tex]\((- \infty, -1.4]\)[/tex].
