Sure! Let's solve and graph the inequality [tex]\( -3x + 4 \leq 7 \)[/tex].
### Step-by-Step Solution
#### Step 1: Isolate the variable [tex]\( x \)[/tex]
To solve the inequality [tex]\( -3x + 4 \leq 7 \)[/tex], we first need to isolate [tex]\( x \)[/tex].
1. Subtract 4 from both sides:
[tex]\[
-3x + 4 - 4 \leq 7 - 4
\][/tex]
Simplifying this, we get:
[tex]\[
-3x \leq 3
\][/tex]
2. Divide both sides by -3. (Remember, when we divide by a negative number, the inequality sign flips):
[tex]\[
x \geq \frac{3}{-3}
\][/tex]
Simplifying this, we get:
[tex]\[
x \geq -1
\][/tex]
Thus, the solution to the inequality is [tex]\( x \geq -1 \)[/tex].
#### Step 2: Express the solution set
In interval notation, the solution set can be written as:
[tex]\[
[-1, \infty)
\][/tex]
### Graphing the Inequality
To graph this inequality, we need to plot the line and the region where the inequality holds.
1. Plot the boundary line: First, graph the line [tex]\( -3x + 4 = 7 \)[/tex]:
[tex]\[
-3x + 4 = 7 \implies -3x = 3 \implies x = -1
\][/tex]
So, the boundary line [tex]\( x = -1 \)[/tex].
2. Shade the region that satisfies the inequality [tex]\( x \geq -1 \)[/tex]: Since [tex]\( x \)[/tex] is greater than or equal to -1, we need to shade the region to the right of the line [tex]\( x = -1 \)[/tex].
Now, let's visualize the graph:
[tex]\[
\begin{array}{cc}
\text{y-axis} & {x \quad \text{axis}} \\
\tikz{\draw[thick] (-3,-1)--(3,-1);} & \tikz{\draw[->] (-3,0) -- (3,0); \draw[->] (0,-3) -- (0,3);\draw[dashed] (-1,-3) -- (-1,3); \draw[fill=gray,fill opacity=0.3] (-1,-3) rectangle (3,3); \draw (1.7,2) node {$ x \geq -1$}; } \\
\end{array}
\][/tex]
- Dashed Line at [tex]\( x = -1 \)[/tex]: The dashed line at [tex]\( x = -1 \)[/tex] indicates the boundary.
- Shaded Region: The region to the right of [tex]\( x = -1 \)[/tex] shows the values of [tex]\( x \)[/tex] that satisfy the inequality [tex]\( -3x + 4 \leq 7 \)[/tex].
Thus, the graph represents [tex]\( x \geq -1 \)[/tex].
### Summary
- Solution: [tex]\( x \geq -1 \)[/tex]
- Interval Notation: [tex]\([-1, \infty)\)[/tex]
- Graph: Shade the region to the right of the line [tex]\( x = -1 \)[/tex].
This concludes our step-by-step solution and graph of [tex]\( -3x + 4 \leq 7 \)[/tex].
