To determine which graph represents the inequality [tex]\( y > -\frac{1}{3}x + 1 \)[/tex], we'll follow a step-by-step approach:
1. Understand the Inequality: The given inequality is [tex]\( y > -\frac{1}{3}x + 1 \)[/tex]. This is the equation of a line with a slope of [tex]\(-\frac{1}{3}\)[/tex] and a y-intercept of [tex]\(1\)[/tex], but it shows that [tex]\( y \)[/tex] should be greater than the line.
2. Graph the Boundary Line: First, we graph the boundary line, [tex]\( y = -\frac{1}{3}x + 1 \)[/tex]. This is a straight line.
- The y-intercept is [tex]\(1\)[/tex], so the line crosses the y-axis at [tex]\( (0, 1) \)[/tex].
- The slope is [tex]\(-\frac{1}{3}\)[/tex], which means for every increase of [tex]\(3\)[/tex] units in [tex]\(x\)[/tex], [tex]\(y\)[/tex] decreases by [tex]\(1\)[/tex] unit. Alternatively, if you increase [tex]\(x\)[/tex] by [tex]\(1\)[/tex] unit, [tex]\(y\)[/tex] decreases by [tex]\(\frac{1}{3}\)[/tex] units.
3. Plot Key Points:
- Start at the y-intercept [tex]\((0,1)\)[/tex].
- From [tex]\((0,1)\)[/tex], move right [tex]\(3\)[/tex] units to [tex]\((3,0)\)[/tex]: [tex]\[\frac{1}{3}\times 3 = 1 \][/tex] so [tex]\( y-1 = 1-1 = 0 \)[/tex].
4. Draw the Line: Connect these points with a straight line. This line will be dashed because the inequality is [tex]\( y > -\frac{1}{3}x + 1 \)[/tex], not [tex]\( y \geq -\frac{1}{3}x + 1 \)[/tex]. The dotted line signifies that points on the line itself are not included in the solution.
5. Shade the Region: Since the inequality is [tex]\( y > -\frac{1}{3}x + 1 \)[/tex], shade the region above the line. This is because this is where [tex]\( y \)[/tex] values are greater than those on the line.
So, the correct graph will show:
- A dashed line representing [tex]\( y = -\frac{1}{3}x + 1 \)[/tex].
- The area above this dashed line shaded to indicate that it includes the region where [tex]\( y \)[/tex] values are greater than those on the line.
- Key points such as [tex]\((0, 1)\)[/tex] and [tex]\((3,0)\)[/tex] helping to define the line, but the region is clearly above this line.
This detailed description should help you identify the correct graph that represents the inequality [tex]\( y > -\frac{1}{3}x + 1 \)[/tex].
