Since no specific problem number was selected, I will solve
Problem 5 as a representative example. The steps below apply to all similar problems on the sheet.
Step 1: Identify the inequalities.
The system given in Problem 5 is:
1) $y \ge \frac{1}{3}x - 1$
2) $y \le \frac{4}{3}x + 2$
Step 2: Analyze the first inequality ($y \ge \frac{1}{3}x - 1$).
*
Boundary Line: The equation is $y = \frac{1}{3}x - 1$.
* The y-intercept is $-1$ (the line crosses the vertical axis at -1).
* The slope is $\frac{1}{3}$ (rise 1, run 3). From $(0, -1)$, go up 1 and right 3 to hit $(3, 0)$.
*
Line Type: Since the symbol is $\ge$ (greater than or equal to), the line is
solid.
*
Shading: Since it is "greater than", we shade
above the line.
Step 3: Analyze the second inequality ($y \le \frac{4}{3}x + 2$).
*
Boundary Line: The equation is $y = \frac{4}{3}x + 2$.
* The y-intercept is $2$ (the line crosses the vertical axis at 2).
* The slope is $\frac{4}{3}$ (rise 4, run 3). From $(0, 2)$, go up 4 and right 3 to hit $(3, 6)$.
*
Line Type: Since the symbol is $\le$ (less than or equal to), the line is
solid.
*
Shading: Since it is "less than", we shade
below the line.
Step 4: Find the Solution Region.
The solution to the system is the area where the shading from Step 2 and Step 3 overlap.
* Looking at the graph for #5, there are two solid red lines.
* One line is less steep (slope $\frac{1}{3}$) and passes through $(0, -1)$. The shading is above this line.
* The other line is steeper (slope $\frac{4}{3}$) and passes through $(0, 2)$. The shading is below this line.
* The overlapping pink region is the wedge shape between these two lines on the right side.
Verification:
Let's test a point inside the shaded region, for example, $(3, 2)$.
* Inequality 1: $2 \ge \frac{1}{3}(3) - 1 \rightarrow 2 \ge 1 - 1 \rightarrow 2 \ge 0$. (True)
* Inequality 2: $2 \le \frac{4}{3}(3) + 2 \rightarrow 2 \le 4 + 2 \rightarrow 2 \le 6$. (True)
Since the point works in both equations, the graph correctly represents the solution.
Final Answer:
The correct graph for Problem 5 shows two solid boundary lines intersecting. The less steep line ($y = \frac{1}{3}x - 1$) has shading above it, and the steeper line ($y = \frac{4}{3}x + 2$) has shading below it. The solution is the overlapping region between these two lines.
Parent Tip: Review the logic above to help your child master the concept of algebra 1 graphing linear inequalities worksheet.
