Graphing Systems of Inequalities Worksheet featuring six systems of inequalities to be graphed on coordinate planes.

Here are the step-by-step solutions for each system of inequalities. To graph these, we first find the boundary line for each inequality and then determine which side to shade.

Problem 1

System:
1) $3x + y \le 2$
2) $6x - 2y > -4$

Step 1: Graph the first inequality ($3x + y \le 2$)
* Rewrite in slope-intercept form ($y = mx + b$): Subtract $3x$ from both sides to get $y \le -3x + 2$.
* Line: Draw a solid line for $y = -3x + 2$. The y-intercept is $(0, 2)$ and the slope is $-3$ (go down 3, right 1). Another point is $(1, -1)$.
* Shading: Since it is $\le$, shade below the line.

Step 2: Graph the second inequality ($6x - 2y > -4$)
* Rewrite in slope-intercept form: Subtract $6x$ to get $-2y > -6x - 4$. Divide by $-2$ and flip the sign: $y < 3x + 2$.
* Line: Draw a dashed line for $y = 3x + 2$. The y-intercept is $(0, 2)$ and the slope is $3$ (go up 3, right 1). Another point is $(1, 5)$.
* Shading: Since it is $<$, shade below the line.

Solution: The solution is the region where the two shaded areas overlap. This looks like a "V" shape opening downwards, with the vertex at $(0, 2)$.

---

Problem 2

System:
1) $6x - 2y \le 4$
2) $y < x$

Step 1: Graph the first inequality ($6x - 2y \le 4$)
* Rewrite: Subtract $6x$ to get $-2y \le -6x + 4$. Divide by $-2$ and flip the sign: $y \ge 3x - 2$.
* Line: Draw a solid line for $y = 3x - 2$. Y-intercept is $(0, -2)$. Slope is $3$. Another point is $(1, 1)$.
* Shading: Since it is $\ge$, shade above the line.

Step 2: Graph the second inequality ($y < x$)
* Line: Draw a dashed line for $y = x$. It passes through $(0,0)$, $(1,1)$, and $(2,2)$.
* Shading: Since it is $<$, shade below the line (towards the bottom right).

Solution: The solution is the wedge-shaped region between the two lines on the right side of the graph. The lines intersect at $(1, 1)$.

---

Problem 3

System:
1) $4x + y < 2$
2) $y > x$

Step 1: Graph the first inequality ($4x + y < 2$)
* Rewrite: $y < -4x + 2$.
* Line: Draw a dashed line for $y = -4x + 2$. Y-intercept is $(0, 2)$. Slope is $-4$ (down 4, right 1). Another point is $(1, -2)$.
* Shading: Shade below the line.

Step 2: Graph the second inequality ($y > x$)
* Line: Draw a dashed line for $y = x$. Passes through $(0,0)$ and $(1,1)$.
* Shading: Shade above the line (towards the top left).

Solution: The solution is the region to the left of the intersection point. The lines intersect where $x = -4x + 2 \rightarrow 5x = 2 \rightarrow x = 0.4$. The overlapping area is the wedge pointing to the left.

---

Problem 4

System:
1) $y < 2x + 1$
2) $y > -\frac{1}{3}x + 4$

Step 1: Graph the first inequality ($y < 2x + 1$)
* Line: Draw a dashed line for $y = 2x + 1$. Y-intercept is $(0, 1)$. Slope is $2$ (up 2, right 1). Point: $(1, 3)$.
* Shading: Shade below the line.

Step 2: Graph the second inequality ($y > -\frac{1}{3}x + 4$)
* Line: Draw a dashed line for $y = -\frac{1}{3}x + 4$. Y-intercept is $(0, 4)$. Slope is $-\frac{1}{3}$ (down 1, right 3). Point: $(3, 3)$.
* Shading: Shade above the line.

Solution: The solution is the triangular region trapped between the two lines on the right side. They intersect at $(1, 3)$.

---

Problem 5

System:
1) $2x - y \ge -6$
2) $x > 2$

Step 1: Graph the first inequality ($2x - y \ge -6$)
* Rewrite: Subtract $2x$ to get $-y \ge -2x - 6$. Divide by $-1$ and flip the sign: $y \le 2x + 6$.
* Line: Draw a solid line for $y = 2x + 6$. Y-intercept is $(0, 6)$. X-intercept is $(-3, 0)$.
* Shading: Shade below the line.

Step 2: Graph the second inequality ($x > 2$)
* Line: Draw a vertical dashed line at $x = 2$.
* Shading: Shade to the right of the line.

Solution: The solution is the area to the right of the vertical line $x=2$ that is also below the diagonal line.

---

Problem 6

System:
1) $y < \frac{1}{2}x + 4$
2) $x - 2y \ge 6$

Step 1: Graph the first inequality ($y < \frac{1}{2}x + 4$)
* Line: Draw a dashed line for $y = \frac{1}{2}x + 4$. Y-intercept is $(0, 4)$. Slope is $\frac{1}{2}$ (up 1, right 2). Point: $(2, 5)$.
* Shading: Shade below the line.

Step 2: Graph the second inequality ($x - 2y \ge 6$)
* Rewrite: Subtract $x$ to get $-2y \ge -x + 6$. Divide by $-2$ and flip the sign: $y \le \frac{1}{2}x - 3$.
* Line: Draw a solid line for $y = \frac{1}{2}x - 3$. Y-intercept is $(0, -3)$. Slope is $\frac{1}{2}$ (up 1, right 2). Point: $(2, -2)$.
* Shading: Shade below the line.

Solution: Notice that both lines have the same slope ($\frac{1}{2}$), so they are parallel. The solution is the entire horizontal strip (band) between the upper dashed line and the lower solid line.

Final Answer:
1. Shade the region below both lines $y = -3x + 2$ (solid) and $y = 3x + 2$ (dashed). The vertex is at $(0,2)$.
2. Shade the region above $y = 3x - 2$ (solid) and below $y = x$ (dashed). Intersection at $(1,1)$.
3. Shade the region below $y = -4x + 2$ (dashed) and above $y = x$ (dashed). Intersection at $(0.4, 0.4)$.
4. Shade the region below $y = 2x + 1$ (dashed) and above $y = -\frac{1}{3}x + 4$ (dashed). Intersection at $(1,3)$.
5. Shade the region below $y = 2x + 6$ (solid) and to the right of $x = 2$ (dashed vertical line).
6. Shade the region between the parallel lines: below $y = \frac{1}{2}x + 4$ (dashed) and below $y = \frac{1}{2}x - 3$ (solid). Effectively, shade the band between the two lines.