Algebra 1 · Unit 4
Show the solution set for a system of two linear inequalities with two variables on a graph.
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≥/≤ solid · >/< dashed
Solution = overlap
Graph each inequality first
Solid or dashed, above or below, just as with one inequality.
The solution is the overlap
A point must make both inequalities true. Shade each half-plane, then keep only where the shading stacks.
Two colors help
One color per inequality. The overlap is the solution region.
Test a point in both
Pick a point in the overlap, such as (0, 0) when it is allowed, and check both original inequalities.
Integer slopes. One solid, one dashed. Graph only in the answer.
Graph each boundary.
First: y ≥ x − 1
Draw y = x − 1 solid.
Shade above.
Second: y < −x + 4
Draw y = −x + 4 dashed.
Shade below.
The solution is the overlap.
Answer: the overlap
Graph each boundary.
First: y ≤ x + 2
Draw y = x + 2 solid.
Shade below.
Second: y ≥ −x + 3
Draw y = −x + 3 solid.
Shade above.
The solution is the overlap.
Answer: the overlap
Graph each boundary.
First: y ≥ 2
Draw y = 2 solid.
Shade above.
Second: y < x − 3
Draw y = x − 3 dashed.
Shade below.
The solution is the overlap.
Answer: the overlap
Graph each boundary.
First: y > 2x − 1
Draw y = 2x − 1 dashed.
Shade above.
Second: y ≤ −x + 5
Draw y = −x + 5 solid.
Shade below.
The solution is the overlap.
Answer: the overlap
Graph each boundary.
First: x ≥ 1
Draw x = 1 solid.
Shade to the right.
Second: y ≤ −x + 2
Draw y = −x + 2 solid.
Shade below.
The solution is the overlap.
Answer: the overlap
Graph each boundary.
First: y < x
Draw y = x dashed.
Shade below.
Second: y < −x + 2
Draw y = −x + 2 dashed.
Shade below.
The solution is the overlap.
Answer: the overlap
Solve for y first. Mixed solid and dashed.
First: y ≤ −x + 5
Solid, shade below.
Second: y ≥ −x + 2
Solid, shade above.
The overlap is the strip between the parallel lines, including the boundaries.
Answer: the overlap
Graph each boundary.
First: y < −2x + 6
Draw y = −2x + 6 dashed.
Shade below.
Second: y ≥ x − 2
Draw y = x − 2 solid.
Shade above.
The solution is the overlap.
Answer: the overlap
Graph each boundary.
First: y ≥ 0.5x − 1
Draw y = 0.5x − 1 solid.
Shade above.
Second: y < −x + 4
Draw y = −x + 4 dashed.
Shade below.
The solution is the overlap.
Answer: the overlap
Graph each boundary.
First: x > −2
Draw x = −2 dashed.
Shade to the right.
Second: y ≤ 3
Draw y = 3 solid.
Shade below.
The solution is the overlap.
Answer: the overlap
−x + y ≥ 2
y ≥ x + 2
Solid, shade above.
y ≤ 2x − 3
Solid, shade below.
Answer: the overlap
First: dashed, shade above y = −2x + 3.
Second: y ≥ −x + 1, solid, shade above.
Answer: the overlap
Flip when dividing by a negative. Steeper lines and mixed open/closed boundaries.
2x − y > 4
− y > −2x + 4
Divide by −1 and flip.
y < 2x − 4
Dashed, shade below.
y ≤ 12x − 2
Solid, shade below.
The solution is the overlap.
Answer: the overlap
Graph each boundary.
First: y ≥ −3x + 5
Draw y = −3x + 5 solid.
Shade above.
Second: y < 0.4x − 1
Draw y = 0.4x − 1 dashed.
Shade below.
The solution is the overlap.
Answer: the overlap
Graph each boundary.
First: x ≤ 2
Draw x = 2 solid.
Shade to the left.
Second: y ≥ 0.6666666666666666x − 1
Draw y = 0.6666666666666666x − 1 solid.
Shade above.
The solution is the overlap.
Answer: the overlap
−2x + y < 3
y < 2x + 3
Dashed, shade below.
y ≥ −1
Solid horizontal, shade above.
Answer: the overlap
Graph each boundary.
First: y ≥ −0.3333333333333333x + 4
Draw y = −0.3333333333333333x + 4 solid.
Shade above.
Second: y < 2x − 3
Draw y = 2x − 3 dashed.
Shade below.
The solution is the overlap.
Answer: the overlap
x − 2y ≥ −2
−2y ≥ −1x − 2
Divide by −2 and flip.
y ≤ 12x + 1
Solid, shade below.
x > −1, dashed vertical, shade right.
Answer: the overlap
The axes give (0, 0).
x = 0 and x + y = 6 → (0, 6).
y = 0 and 2x + y = 8 → (4, 0).
x + y = 6 and 2x + y = 8.
Subtract: x = 2, so y = 4.
Corner points: (0, 0), (0, 6), (2, 4), (4, 0).
Answer: (0, 0), (0, 6), (2, 4), (4, 0)
Both boundaries have slope 1.
y ≥ x is on or above y = x.
y ≤ x − 2 is on or below a parallel line two units down.
The half-planes do not overlap, so there is no solution.
Answer: no solution
Graph each boundary.
First: y > 2
Draw y = 2 dashed.
Shade above.
Second: x ≤ −1
Draw x = −1 solid.
Shade to the left.
The solution is the overlap.
(−3, 4) is in the overlap: 4 > 2 and −3 ≤ −1.
(0, 4) has y > 2 true but x ≤ −1 false.
(−3, 0) has x ≤ −1 true but y > 2 false.
Answer: overlap e.g. (−3, 4)
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