Certainly! Let's solve the inequality step-by-step and then graph the solution on a number line.
The given inequality is:
[tex]\[ -4x + 7 > x - 13 \][/tex]
Step 1: Move all terms involving [tex]\( x \)[/tex] to one side.
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ -4x + 7 - x > x - 13 - x \][/tex]
[tex]\[ -5x + 7 > -13 \][/tex]
Step 2: Move constant terms to the other side of the inequality.
Subtract 7 from both sides:
[tex]\[ -5x + 7 - 7 > -13 - 7 \][/tex]
[tex]\[ -5x > -20 \][/tex]
Step 3: Isolate [tex]\( x \)[/tex].
Divide both sides by [tex]\( -5 \)[/tex]. Remember that dividing by a negative number reverses the inequality sign:
[tex]\[ x < \frac{-20}{-5} \][/tex]
[tex]\[ x < 4 \][/tex]
This means [tex]\( x \)[/tex] must be less than 4.
So, the solution set in interval notation is:
[tex]\[ (-\infty, 4) \][/tex]
Graphing on Number Line:
- Draw a number line.
- Mark the point [tex]\( 4 \)[/tex] on the number line.
- Use an open circle at [tex]\( 4 \)[/tex] to indicate that [tex]\( 4 \)[/tex] is not included in the solution.
- Shade the region to the left of 4 to indicate all numbers less than 4.
Here is the graphical representation on a number line:
[tex]\[ \begin{array}{cccccccccccccccc}
-\infty & \text{<} & \cdots & -3 & \text{<} & -2 & \text{<} & -1 & \text{<} & 0 & \text{<} & 1 & \text{<} & 2 & \text{<} & 3 & \text{<} & 4 \\
& & & & & & & & & & & & & & \circ & \\
\end{array} \][/tex]
The open circle at 4 and shading to the left represents all [tex]\( x \)[/tex] values less than 4.
