Graphing Inequalities in Two Variables

Graphing inequalities in two variables. In MathBound, the free Algebra I adventure game, Shade Painter asks these problems, and the only way past is to solve them.

Part of Systems of Equations and Inequalities: chapter 6 of the standard course.

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10 questions, no sign-up, nothing saved. Climbs from the basics to harder questions.

What you will practice

Verify Solutions.
Graph Boundary Lines. Identify and graph a solid boundary line for inclusive inequalities (<= or >=).
Graph Inequalities. Graph an inequality by testing a point not on the line (usually (0,0)) and shading the correct region.
Horizontal and Vertical Inequalities. Graph a strict or inclusive horizontal or vertical inequality.

Worked examples

Made by the game’s own question engine. Try each one, then open its steps.

Example 1 · Verify Solutions

Which

ordered

pair

is

a

solution

to

3 x + 3 y \leq 4 ?

$(0, 1)$
$(3, 2)$
$(4, 2)$
$(- 1, 4)$

Show the worked steps

$Substitute:$ $3 (0) + 3 (1) = 3 .$
$3 \leq 4,$ $so$ $(0, 1)$ $is$ $a$ $solution.$

Answer: $(0, 1)$

Example 2 · Graph Boundary Lines

What

boundary

line

does

5 x - 1 \leq y

have?

$A$ $dashed$ $line$ $at$ $y = 5 x - 1$
$A$ $solid$ $line$ $at$ $y = 5 x + 1$
$A$ $solid$ $line$ $at$ $y = 5 x - 1$
$A$ $solid$ $line$ $at$ $y = - 5 x - 1$

Show the worked steps

$Which$ $side$ $y$ $is$ $on$ $does$ $not$ $change$ $the$ $boundary:$ $it$ $is$ $y = 5 x - 1 .$
$The$ $symbol$ $includes$ $the$ $boundary,$ $so$ $the$ $line$ $is$ $solid.$

Answer: $A$ $solid$ $line$ $at$ $y = 5 x - 1$

Example 3 · Graph Inequalities

How

is

y > x + 5

graphed?

$A$ $dashed$ $line$ $at$ $y = x + 5,$ $shaded$ $below$
$A$ $solid$ $line$ $at$ $y = x + 5,$ $shaded$ $above$
$A$ $solid$ $line$ $at$ $y = x + 5,$ $shaded$ $below$
$A$ $dashed$ $line$ $at$ $y = x + 5,$ $shaded$ $above$

Show the worked steps

$>$ $is$ $strict,$ $so$ $the$ $line$ $is$ $dashed.$
$Test$ $(0, 0) : 0 > 5$ $is$ $false,$ $so$ $shade$ $above$ $the$ $line.$

Answer: $A$ $dashed$ $line$ $at$ $y = x + 5,$ $shaded$ $above$

Where it is in your textbook

Textbook	Lessons
Standard course	6.5, 6.6
Big Ideas Math Algebra 1	5.6, 5.7
Big Ideas Math Algebra 1 (2022)	5.6, 5.7
Carnegie Learning Algebra I	M2 3.3, M2 3.4
CPM Core Connections Algebra	9.3.1, 9.3.2, 9.4.1, 9.4.2
CPS Skyline Algebra	5.2, 5.4, 5.5-5.6
enVision Algebra 1	4-4, 4-5
enVision Algebra 1 (2024)	4-4, 4-5
Eureka Math Algebra I (EngageNY)	1.20-1.21, 1.22-1.23
Glencoe Algebra 1	5-6, 6-6
HMH Into Algebra 1	10.1, 10.2
Holt McDougal Larson Algebra 1	5.7, 6.6
Illustrative Mathematics Algebra 1	2.21-2.23, 2.24-2.26
Illustrative Mathematics Algebra 1 (v.360)	4.4-4.6, 4.7-4.9
Math Nation Algebra 1 (Florida B.E.S.T.)	6.4, 6.5, 6.6, 6.7, 6.9, 6.10
MVP Mathematics I (Integrated Math 1)	5.2, 5.3, 5.5, 5.6, 5.7
Open Up NC Math 1 (MVP)	5.2, 5.3, 5.5, 5.6, 5.7
OpenStax Elementary Algebra 2e	4.7, 5.6
Pearson Algebra 1 Common Core	6-5, 6-6
Reveal Algebra 1	6-5, 7-5
SpringBoard Algebra 1	16, 18

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