Sure! Let's solve and graph the inequality [tex]\(-3x + 4 \leq 7\)[/tex] step by step.
### Step-by-Step Solution
1. Start with the given inequality:
[tex]\[
-3x + 4 \leq 7
\][/tex]
2. Isolate the variable term [tex]\(-3x\)[/tex]:
Subtract 4 from both sides of the inequality:
[tex]\[
-3x + 4 - 4 \leq 7 - 4
\][/tex]
Simplifying this:
[tex]\[
-3x \leq 3
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Divide both sides by [tex]\(-3\)[/tex]. Since we are dividing by a negative number, we need to reverse the inequality sign:
[tex]\[
x \geq \frac{3}{-3}
\][/tex]
Simplifying this:
[tex]\[
x \geq -1
\][/tex]
### Solution in Interval Notation
The solution set can be expressed in interval notation as:
[tex]\[
[-1, \infty)
\][/tex]
### Graphing the Inequality
To graph the solution [tex]\([-1, \infty)\)[/tex]:
1. Draw a number line.
2. Locate the point [tex]\(-1\)[/tex] on the number line.
3. Since the inequality is [tex]\(\geq -1\)[/tex], place a closed circle at [tex]\(-1\)[/tex] to indicate that [tex]\(-1\)[/tex] is included in the solution.
4. Shade the number line to the right of [tex]\(-1\)[/tex], extending to positive infinity. This shading represents all numbers greater than or equal to [tex]\(-1\)[/tex].
Here's a visual representation of the graph:
```
<----|----|----|----|----|----|----|----|------>
-3 -2 -1 0 1 2 3 4 5
[=======================>
```
The closed circle at [tex]\(-1\)[/tex] indicates that [tex]\(-1\)[/tex] is included, and the shading to the right shows that all values greater than [tex]\(-1\)[/tex] are also part of the solution.
So, the solution to the inequality [tex]\(-3x + 4 \leq 7\)[/tex] is:
[tex]\[
x \geq -1
\][/tex]
And in interval notation, it is:
[tex]\[
[-1, \infty)
\][/tex]
