To solve the inequality and determine which graph represents [tex]\( y \leq \frac{2}{2} x - 1 \)[/tex], let's break down the process step by step.
### Step 1: Simplify the Inequality
The given inequality is:
[tex]\[ y \leq \frac{2}{2} x - 1 \][/tex]
First, simplify the coefficient of [tex]\( x \)[/tex]:
[tex]\[ \frac{2}{2} = 1 \][/tex]
So, the inequality simplifies to:
[tex]\[ y \leq x - 1 \][/tex]
### Step 2: Identify the Boundary Line
The boundary line for the inequality [tex]\( y \leq x - 1 \)[/tex] is:
[tex]\[ y = x - 1 \][/tex]
### Step 3: Plot the Boundary Line
To draw the line [tex]\( y = x - 1 \)[/tex], identify two points on the line:
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 0 - 1 = -1 \][/tex]
Point: [tex]\( (0, -1) \)[/tex]
2. For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 1 - 1 = 0 \][/tex]
Point: [tex]\( (1, 0) \)[/tex]
Plot these points and draw a straight line through them.
### Step 4: Determine the Shaded Region
Since the inequality is [tex]\( y \leq x - 1 \)[/tex], the region that satisfies this inequality is below or on the line [tex]\( y = x - 1 \)[/tex].
### Summary
The graph you should be looking for is one that has:
- A line passing through [tex]\((0, -1)\)[/tex] and [tex]\((1, 0)\)[/tex].
- The region below and including this line shaded.
Thus, the correct graph is the one where the region below the line [tex]\( y = x - 1 \)[/tex] is shaded, indicating that all the points [tex]\( (x, y) \)[/tex] on and below this line satisfy the inequality [tex]\( y \leq x - 1 \)[/tex].
Since the solution is 1, the correct graph is the first one among the options provided.
