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GRAPHING LINEAR EQUATIONS IN TWO VARIABLES
To graph a linear inequality with one variable we use the number line. To graph a linear inequality
with two variables we use a two-dimensional graph. As with linear equations, linear inequalities can be
written in standard form or slope intercept form.
Standard Form:
Ax + By < C or Ax + By > C
Ax + By ≤ C or Ax + By ≥ C
y < mx + b or y > mx + b
y ≤ mx + b or y ≥ mx + b
Slope-Intercept Form:
To graph a linear inequality with two variables it is probably easier and recommended to change it to
slope intercept form. There are two important rules to remember when graphing a linear inequality. If
the inequality sign is > or <, the line is dashed. If the inequality is ≥ or ≤ , then the line is solid. Also,
for inequalities that are greater than > or greater or equal to ≥ the graph needs to be shaded above the
line. For inequalities that are less than < or less or equal to ≤ the graph needs to be shaded below the
line.
Examples:
Graph the solutions of 3 x + 2 y ≥ 6 :
Step 1 – Since the equation is in standard form, change it to slope intercept form.
3x + 2 y ≥ 6
→
2 y ≥ −3 x + 6
−3 x + 6
y≥
2
3
y ≥ − x+3
2
← Slope-Intercept Form
Step 2 – Graph the slope-intercept form equation. Since the inequality has the
sign ≥ the line is solid.
Step 3 – Since the equation has the sign ≥ , the graph is shaded above the line.
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← The shaded area in the
graph is the graphic
answer to the inequality.
It means that any point
above the line, including
the line, satisfies the
inequality and can be a
possible answer.
Graph the solutions of x − 3 y < 4
Step 1 – Since the equation is in standard form, change it to slope intercept form.
x − 3y < 4
→
− 3y < −x + 4
−x + 4
y>
−3
Remember to switch the
inequality sign when
dividing by a negative.
y>
1
4
x−
3
3
← Slope-Intercept Form
Step 2 – Graph the slope-intercept form equation. Since the inequality has the
sign > the line is dashed.
Step 3 – Since the equation has the sign >, the graph is shaded above the line.
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Solving a System of Linear Inequalities
To solve a system of linear inequalities we need to do it graphically since the solution to a system of
linear inequalities is the set of points whose coordinates satisfy all the inequalities in the system, or
where both shaded areas overlap. Graph the lines as previously shown.
1
y <− x+2
Example:
Determine the solution to the following system of inequalities
2
x− y ≤4
Step 1 – Change the second inequality to slope-intercept form.
x− y≤4
x− x− y ≤ 4− x
− y ≤ −x + 4
y ≥ x−4
← Remember to switch the inequality sign since
we are dividing by a negative number.
Step 2 – Graph both inequalities in slope-intercept form and shade the solution
for each inequality. Since the first inequality has the sign < the line is
dashed. The second inequality has the sign ≥ so the line is solid.
y ≥ x−4
Solution to the
system of
inequalities
y<−
5
1
x+2
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GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES – EXERCISES
Graph each inequality:
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1. y ≥ − x
2
2. y < 2 x + 1
3. 10 ≥ 5 x − 2 y
Determine the solution to each system of inequalities:
2x − y < 4
y ≥ −x + 2
5.
y < 3x − 4
6x ≥ 2 y + 8
x≥0
1
1
7. x + y ≥ 2
2
2
2 x − 3 y ≤ −6
x≥0
y≥0
8.
2x + 3y ≤ 6
4x + y ≤ 4
x≥0
y≥0
9.
5 x + 4 y ≤ 16
x + 6 y ≤ 18
4.
6.
− 3 x + 2 y ≥ −5
y ≤ −4 x + 7
x≥0
y≥0
10. x ≤ 15
30 x + 25 y ≤ 750
10 x + 40 y ≤ 800
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GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES – ANSWERS TO EXERCISES
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