Graphing systems of linear inequalities worksheet with solutions.

To solve the problems shown in the worksheet, we need to graph systems of linear inequalities. The solution is the region where the shaded areas of all inequalities in a system overlap.

Here is the step-by-step solution for each problem visible on the sheet.

Problem 1

Inequalities:
1. $y \leq -\frac{3}{4}x + 3$
2. $y > \frac{1}{2}x - 1$

Step 1: Graph the first inequality ($y \leq -\frac{3}{4}x + 3$)
* Y-intercept: Start at $(0, 3)$.
* Slope: Go down 3 units and right 4 units to find another point $(4, 0)$.
* Line Type: Since it is $\leq$, draw a solid line.
* Shading: Test a point like $(0,0)$. $0 \leq 3$ is true, so shade below the line (towards the origin).

Step 2: Graph the second inequality ($y > \frac{1}{2}x - 1$)
* Y-intercept: Start at $(0, -1)$.
* Slope: Go up 1 unit and right 2 units to find another point $(2, 0)$.
* Line Type: Since it is $>$, draw a dashed line.
* Shading: Test $(0,0)$. $0 > -1$ is true, so shade above the line.

Solution Region: The answer is the double-shaded area that is below the solid line and above the dashed line.

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Problem 2

Inequalities:
1. $y < 2x + 4$
2. $y \geq -x - 2$

Step 1: Graph the first inequality ($y < 2x + 4$)
* Y-intercept: Start at $(0, 4)$.
* Slope: Go up 2 units and right 1 unit.
* Line Type: Dashed line (because of $<$).
* Shading: Shade below the line.

Step 2: Graph the second inequality ($y \geq -x - 2$)
* Y-intercept: Start at $(0, -2)$.
* Slope: Go down 1 unit and right 1 unit.
* Line Type: Solid line (because of $\geq$).
* Shading: Shade above the line.

Solution Region: The overlapping area between the two lines.

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Problem 3

Inequalities:
1. $y \leq x - 3$
2. $y > -2x + 6$

Step 1: Graph the first inequality ($y \leq x - 3$)
* Y-intercept: Start at $(0, -3)$.
* Slope: Go up 1 unit and right 1 unit.
* Line Type: Solid line.
* Shading: Shade below the line.

Step 2: Graph the second inequality ($y > -2x + 6$)
* Y-intercept: Start at $(0, 6)$.
* Slope: Go down 2 units and right 1 unit.
* Line Type: Dashed line.
* Shading: Shade above the line.

Solution Region: The V-shaped region opening to the right where the shadings overlap.

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Problem 4

Inequalities:
1. $x + y \leq 4$ $\rightarrow$ Rewrite as $y \leq -x + 4$
2. $x - y \leq 2$ $\rightarrow$ Rewrite as $-y \leq -x + 2$ $\rightarrow$ $y \geq x - 2$ (Flip the sign when dividing by negative)

Step 1: Graph $y \leq -x + 4$
* Intercepts: $(0, 4)$ and $(4, 0)$.
* Line Type: Solid.
* Shading: Below the line.

Step 2: Graph $y \geq x - 2$
* Intercepts: $(0, -2)$ and $(2, 0)$.
* Line Type: Solid.
* Shading: Above the line.

Solution Region: The intersection of the two half-planes.

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Problem 5

Inequalities:
1. $3x + 2y \leq 12$
2. $x - 2y \geq 2$

Step 1: Graph $3x + 2y \leq 12$
* Find intercepts:
* If $x=0$, $2y=12 \rightarrow y=6$. Point: $(0,6)$.
* If $y=0$, $3x=12 \rightarrow x=4$. Point: $(4,0)$.
* Line Type: Solid.
* Shading: Test $(0,0)$. $0 \leq 12$ is true. Shade towards the origin (below/left).

Step 2: Graph $x - 2y \geq 2$
* Find intercepts:
* If $x=0$, $-2y=2 \rightarrow y=-1$. Point: $(0,-1)$.
* If $y=0$, $x=2$. Point: $(2,0)$.
* Line Type: Solid.
* Shading: Test $(0,0)$. $0 \geq 2$ is false. Shade away from the origin (below/right).

Solution Region: The wedge shape formed by the overlap below both lines.

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Problem 6

Inequalities:
1. $2x + 3y \geq 6$
2. $x - y \leq 1$

Step 1: Graph $2x + 3y \geq 6$
* Find intercepts:
* If $x=0$, $3y=6 \rightarrow y=2$. Point: $(0,2)$.
* If $y=0$, $2x=6 \rightarrow x=3$. Point: $(3,0)$.
* Line Type: Solid.
* Shading: Test $(0,0)$. $0 \geq 6$ is false. Shade away from the origin (above/right).

Step 2: Graph $x - y \leq 1$
* Rewrite as $y \geq x - 1$.
* Find intercepts:
* If $x=0$, $y=-1$. Point: $(0,-1)$.
* If $y=0$, $x=1$. Point: $(1,0)$.
* Line Type: Solid.
* Shading: Since it is $y \geq$, shade above the line.

Solution Region: The area above both lines.

Final Answer:
The solutions are the graphed regions described above. For each problem, the correct graph shows the specific overlapping shaded area bounded by the calculated lines (solid or dashed depending on the inequality symbol).