Sure, let's solve the inequality step-by-step and illustrate the solution on a number line.
### Step-by-Step Solution
1. Start with the given inequality:
[tex]\[
4x - 13 \leq 15
\][/tex]
2. Isolate the term involving [tex]\( x \)[/tex]:
Add 13 to both sides of the inequality to get rid of the constant term on the left side:
[tex]\[
4x - 13 + 13 \leq 15 + 13
\][/tex]
Simplifying this, we obtain:
[tex]\[
4x \leq 28
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Divide both sides by 4 to isolate [tex]\( x \)[/tex]:
[tex]\[
\frac{4x}{4} \leq \frac{28}{4}
\][/tex]
Simplifying this, we get:
[tex]\[
x \leq 7
\][/tex]
### Solution in Interval Notation
The solution to the inequality [tex]\( 4x - 13 \leq 15 \)[/tex] is [tex]\( x \leq 7 \)[/tex]. In interval notation, this can be written as:
[tex]\[
(-\infty, 7]
\][/tex]
### Number Line Representation
To represent [tex]\( x \leq 7 \)[/tex] on a number line:
1. Draw a number line.
2. Mark the point [tex]\( 7 \)[/tex] on the number line.
3. Shade the region to the left of [tex]\( 7 \)[/tex], including the point [tex]\( 7 \)[/tex] itself (indicating that the solution includes all numbers less than or equal to [tex]\( 7 \)[/tex]).
The number line is illustrated as:
```
<---|---|---|---|---|---|---|---|---|---|---|---|---|---|--->
-3 -2 -1 0 1 2 3 4 5 6 [7] 8 9 10
]
```
The shaded part covers all numbers to the left of 7, including 7 itself, indicating that all these numbers satisfy the inequality [tex]\( x \leq 7 \)[/tex].
