1. Inequalities in Two Variables
4.4
Inequalities in Two Variables
Graphs of Linear Inequalities
Systems of Linear Inequalities
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2. Graphs of Linear Inequalities
When the equal sign in a linear equation is replaced with an inequality sign, a linear inequality is formed. Solutions of linear inequalities are ordered pairs.
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3. Example
Determine whether (1, 5) and (6, –2) are solutions of the inequality 3x – y < 5.
Solution
3(1) –
–2 < 5
TRUE
3x – y < 5
3(6) – (–2) 5
20 < 5
FALSE
3x – y < 5
The pair (1, 5) is a solution of the inequality, but (6, –2) is not.
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4. The graph of a linear equation is a straight line
The graph of a linear equation is a straight line. The graph of a linear inequality is a half-plane, with a boundary that is a straight line. To find the equation of the boundary line, we simply replace the inequality sign with an equals sign.
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5. Example Solution Graph
First graph the boundary y = x. Since the inequality is greater than or equal to, the line is drawn solid and is part of the graph of x y -3 2 -2 3 -1 1 6 5 4
y = x -4 -5
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6. Solution
(continued)
Note that in the graph on the right each ordered pair on the half-plane above y = x contains a y x -3 2 -2 3 -1 1 6 5 4
y = x -4 -5
y- coordinate that is greater than the x- coordinate It turns out that any point on the same side as (–2, 2) is also a solution Thus, if one point in a half- plane is a solution, then all points in that half-plane are solutions.
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7. Solution (continued)
We finish drawing the solution set by shading the half-plane above y = x. The complete solution
set consists of the shaded half-plane as well as the boundary itself. x y -3 2 -2 3 -1 1 6 5 4
y = x -4 -5
For any point here, y > x.
For any point here, y = x.
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8. Example Solution Graph (3, 1)
Since the inequality sign is < , points on the line y = 3 – 8x do not represent solutions of the inequality, so the line is dashed Using (3, 1) as a test point, we see that it is not a solution. Thus the points in the other half-plane are solutions. x y -3 2 -2 3 -1 1 6 5 4
y = 3 – 8x -4 -5
(3, 1)
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9. Steps for Graphing Linear Inequalities
Replace the inequality sign with an equals sign and graph this line as the boundary. If the inequality symbol is < or >, draw the line dashed. If the inequality symbol is , draw the line solid.
The graph of the inequality consists of a half-plane on one side of the line and, if the line is solid, the line as well.
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10. Steps for Graphing Linear Inequalities
a) If the inequality is of the form y < mx + b or shade below the line.
If the inequality is of the form y > mx + b or
shade above the line.
b) If y is not isolated, either solve for y and graph as in part (a) or simply graph the boundary and use a test point. If the test point is a solution, shade the half-plane containing the point. If it is not a solution, shade the other half-plane.
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11. Example Solution Graphy -3 2 -2 3 -1 1 6 5 4
y = (1/6)x – 1 -4 -5
The graph consists of the half-plane above the dashed boundary line y = (1/6)x – 1.
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12. Example Solution Graphy -3 2 -2 3 -1 1 6 5 4 -4 -5
The graph consists of the line x = –3 and the half-plane to the right of the line x = –3.
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13. Systems of Linear Inequalities
To graph a system of equations, we graph the individual equations and then find the intersection of the individual graphs. We do the same thing for a system of inequalities, that is, we graph each inequality and find the intersection of the individual graphs.
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14. Example Solution First graph x + y > 3 in red. Graph the system y x -3 2 -2 3 -1 1 6 5 4 -4 -5
First graph x + y > 3 in red.
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15. Solution Next graph in blue. (continued) y x 6 5 4 3 2 1 -3 2 -2 3 -1 1 6 5 4 -4 -5
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16. Solution Now we find the intersection of the regions. (continued)x y -3 2 -2 3 -1 1 6 5 4 -4 -5
Solution set to the system
Now we find the intersection of the regions.
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17. Example Solution This is a system of inequalities Graph Red Blue -3 2 -2 3 -1 1 6 5 4 -4 -5
Solution set
This is a system of inequalities
Red
Blue
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18. A system of inequalities may have a graph that consists of a polygon and its interior. In Section 4.5 we will have use for the corners, or vertices (singular vertex), of such a graph.
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19. Example The graph of the system Blue Green (3, 5) Red -3 2 -2 3 -1 1 6 5 4 -4 -5
(3, 5)
(3, –3 )
(–1, 1 )
is shown below and the vertices are labeled. These vertices can be found by finding the intersection points of the pairs of lines.
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20. Linear equations 2x – 8 = 3(x + 5) A number in one variable
Let’s look at 6 different types of problems that we have solved, along with illustrations of each type.
Type Example Solution
Linear equations 2x – 8 = 3(x + 5) A number
in one variable
Graph
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21. Linear inequalities –3x + 5 > 2 A set of numbers;
Type Example Solution
Linear inequalities –3x + 5 > A set of numbers;
in one variable an interval
Graph
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22. Linear equations 2x + y = 7 A set of ordered
Type Example Solution
Linear equations 2x + y = A set of ordered
in two variables pairs; a line
Graph
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23. Linear inequalities x + y ≥ 4 A set of ordered
Type Example Solution
Linear inequalities x + y ≥ A set of ordered
in two variables pairs; a half-plane
Graph
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24. System of x + y = 3, An ordered pair or
Type Example Solution
System of x + y = 3, An ordered pair or
equations in x – y = – a (possibly empty)
two variables set of ordered pairs
Graph
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25. two variables x + 7 ≥ 0 of a plane
Type Example Solution
System of x – 2y ≤ 12, A set of ordered inequalities in y – 3 ≤ 0, pairs; a region
two variables x + 7 ≥ of a plane
Graph
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