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Math Coach Amy
Amy Ferguson Moncure

H Algebra 2/Trig · Unit 1

Systems of Linear Inequalities

Graph two linear inequalities on the same plane. Solid or dashed, above or below, and keep the overlap.

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Key Ideas Easy Medium Hard

≥/≤ solid  ·  >/< dashed

Solution = overlap

What to Remember

Solid or dashed

≥ or ≤ is a solid boundary (points on the line count). > or < is dashed (points on the line do not count).

Above or below

y > mx + c shades above the line. y < mx + c shades below. A test point such as (0, 0) checks the side when the line does not go through the origin.

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.

Level 1

Easy

Integer slopes. One solid, one dashed.

Example
Graph the system.y ≥ x − 1y < −x + 4

Graph each boundary.

First: y ≥ x − 1

Draw y = x − 1 solid.

Shade above the line.

Second: y < −x + 4

Draw y = −x + 4 dashed.

Shade below the line.

The solution is the overlap.

Answer: the overlap

1
Graph the system.y ≤ x + 2y ≥ −x + 3
Show Answer

Graph each boundary.

First: y ≤ x + 2

Draw y = x + 2 solid.

Shade below the line.

Second: y ≥ −x + 3

Draw y = −x + 3 solid.

Shade above the line.

The solution is the overlap.

Answer: the overlap

2
Graph the system.y ≥ 2y < x − 3
Show Answer

Graph each boundary.

First: y ≥ 2

Draw y = 2 solid.

Shade above the line.

Second: y < x − 3

Draw y = x − 3 dashed.

Shade below the line.

The solution is the overlap.

Answer: the overlap

3
Graph the system.y > 2x − 1y ≤ −x + 5
Show Answer

Graph each boundary.

First: y > 2x − 1

Draw y = 2x − 1 dashed.

Shade above the line.

Second: y ≤ −x + 5

Draw y = −x + 5 solid.

Shade below the line.

The solution is the overlap.

Answer: the overlap

4
Graph the system.y ≥ −2x + 3y ≥ x − 4
Show Answer

Graph each boundary.

First: y ≥ −2x + 3

Draw y = −2x + 3 solid.

Shade above the line.

Second: y ≥ x − 4

Draw y = x − 4 solid.

Shade above the line.

The solution is the overlap.

Answer: the overlap

5
Graph the system.y < xy < −x + 2
Show Answer

Graph each boundary.

First: y < x

Draw y = x dashed.

Shade below the line.

Second: y < −x + 2

Draw y = −x + 2 dashed.

Shade below the line.

The solution is the overlap.

Answer: the overlap

Level 2

Medium

Fractional slopes. Mixed solid and dashed.

Example
Graph the system.y ≥ −x + 3y < 12x − 2

Graph each boundary.

First: y ≥ −x + 3

Draw y = −x + 3 solid.

Shade above the line.

Second: y < 12x − 2

Draw y = 12x − 2 dashed.

Shade below the line.

The solution is the overlap.

Answer: the overlap

1
Graph the system.y ≤ 12x + 1y > −2x + 4
Show Answer

Graph each boundary.

First: y ≤ 12x + 1

Draw y = 12x + 1 solid.

Shade below the line.

Second: y > −2x + 4

Draw y = −2x + 4 dashed.

Shade above the line.

The solution is the overlap.

Answer: the overlap

2
Graph the system.y ≥ 23x − 1y ≥ −12x + 3
Show Answer

Graph each boundary.

First: y ≥ 23x − 1

Draw y = 23x − 1 solid.

Shade above the line.

Second: y ≥ −12x + 3

Draw y = −12x + 3 solid.

Shade above the line.

The solution is the overlap.

Answer: the overlap

3
Graph the system.y < −32x + 2y ≤ 13x − 1
Show Answer

Graph each boundary.

First: y < −32x + 2

Draw y = −32x + 2 dashed.

Shade below the line.

Second: y ≤ 13x − 1

Draw y = 13x − 1 solid.

Shade below the line.

The solution is the overlap.

Answer: the overlap

4
Graph the system.y > 32x − 4y ≥ −x + 1
Show Answer

Graph each boundary.

First: y > 32x − 4

Draw y = 32x − 4 dashed.

Shade above the line.

Second: y ≥ −x + 1

Draw y = −x + 1 solid.

Shade above the line.

The solution is the overlap.

Answer: the overlap

5
Graph the system.y ≤ −13x + 4y < 2x − 3
Show Answer

Graph each boundary.

First: y ≤ −13x + 4

Draw y = −13x + 4 solid.

Shade below the line.

Second: y < 2x − 3

Draw y = 2x − 3 dashed.

Shade below the line.

The solution is the overlap.

Answer: the overlap

Level 3

Hard

Steeper lines and mixed open/closed boundaries.

Example
Graph the system.y ≥ −3x + 5y < 25x − 3

Graph each boundary.

First: y ≥ −3x + 5

Draw y = −3x + 5 solid.

Shade above the line.

Second: y < 25x − 3

Draw y = 25x − 3 dashed.

Shade below the line.

The solution is the overlap.

Answer: the overlap

1
Graph the system.y < −14x + 6y ≥ 32x − 2
Show Answer

Graph each boundary.

First: y < −14x + 6

Draw y = −14x + 6 dashed.

Shade below the line.

Second: y ≥ 32x − 2

Draw y = 32x − 2 solid.

Shade above the line.

The solution is the overlap.

Answer: the overlap

2
Graph the system.y ≤ 4x − 1y ≤ −15x + 2
Show Answer

Graph each boundary.

First: y ≤ 4x − 1

Draw y = 4x − 1 solid.

Shade below the line.

Second: y ≤ −15x + 2

Draw y = −15x + 2 solid.

Shade below the line.

The solution is the overlap.

Answer: the overlap

3
Graph the system.y > −52x + 1y < 12x + 3
Show Answer

Graph each boundary.

First: y > −52x + 1

Draw y = −52x + 1 dashed.

Shade above the line.

Second: y < 12x + 3

Draw y = 12x + 3 dashed.

Shade below the line.

The solution is the overlap.

Answer: the overlap

4
Graph the system.y ≥ 15x − 4y > −32x + 2
Show Answer

Graph each boundary.

First: y ≥ 15x − 4

Draw y = 15x − 4 solid.

Shade above the line.

Second: y > −32x + 2

Draw y = −32x + 2 dashed.

Shade above the line.

The solution is the overlap.

Answer: the overlap

5
Graph the system.y ≤ −2x + 6y ≥ 34x − 1
Show Answer

Graph each boundary.

First: y ≤ −2x + 6

Draw y = −2x + 6 solid.

Shade below the line.

Second: y ≥ 34x − 1

Draw y = 34x − 1 solid.

Shade above the line.

The solution is the overlap.

Answer: the overlap

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