To solve the given inequality [tex]\(\frac{x}{5} - 4 > -3\)[/tex] for [tex]\(x\)[/tex], we will follow a step-by-step approach:
### Step 1: Isolate the variable term
First, we need to get the term with [tex]\(x\)[/tex] on one side by itself. We start by adding 4 to both sides of the inequality:
[tex]\[
\frac{x}{5} - 4 + 4 > -3 + 4
\][/tex]
[tex]\[
\frac{x}{5} > 1
\][/tex]
### Step 2: Eliminate the fraction
Next, we need to eliminate the fraction by multiplying both sides of the inequality by 5:
[tex]\[
5 \cdot \frac{x}{5} > 5 \cdot 1
\][/tex]
[tex]\[
x > 5
\][/tex]
### Step 3: Interpret the solution
The solution to the inequality [tex]\(x > 5\)[/tex] is all real numbers greater than 5. In interval notation, this can be expressed as:
[tex]\[
(5, \infty)
\][/tex]
### Step 4: Graph the solution set
To graph the solution set [tex]\(x > 5\)[/tex], we can follow these steps:
1. Draw a number line.
2. Find and mark the point 5 on the number line.
3. Since the inequality is strict (i.e., [tex]\(>\)[/tex] not [tex]\(\geq\)[/tex]), we use an open circle at 5 to indicate that 5 is not included in the solution set.
4. Shade the number line to the right of 5 to represent all numbers greater than 5.
Here is the visual representation of the solution on the number line:
```
<--|----|----|----|----|----|----|----|----|----|-->
0 1 2 3 4 5 6 7 8 ...
o======>
```
The shaded part and the open circle at 5 represent the solutions for [tex]\(x > 5\)[/tex].
So, the inequality [tex]\(\frac{x}{5} - 4 > -3\)[/tex] has [tex]\(x > 5\)[/tex] as its solution, and this is shown on the graph as described.
