Unit 4, Lesson 4
Graphing Linear Inequalities in Two Variables (Part 1)
Algebra 1
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Here are four inequalities. Study each inequality assigned to your group and work with your group to:
Here is a graph that represents solutions to the equation .
Sketch 4 quick graphs representing the solutions to each of these inequalities:
AInequality graphed on a coordinate plane, origin O. Each axis from negative 10 to 8, by 2’s. Dashed line passes through 3 comma 8, 3 comma 0, and 3 comma negative 10. The region to the right of the dashed line is shaded.
BInequality graphed on a coordinate plane, origin O. Each axis from negative 10 to 8, by 2’s. Solid line passes through negative 10 comma negative 2, negative 2 comma negative 2, and 10 comma negative 2. The region below the solid line is shaded.
CInequality graphed on a coordinate plane, origin O. Each axis from negative 10 to 8, by 2’s. Dashed line passes through negative 10 comma negative 10, 0 comma 0, and 10 comma 10. The region above the dashed line is shaded.
DInequality graphed on a coordinate plane, origin O. Each axis from negative 10 to 8, by 2’s. Solid line passes through negative 10 comma negative 10, 0 comma 0, and 10 comma 10. The region below the solid line is shaded.
The equation is an equation in two variables. Its solution is any pair of and whose sum is 7. The pairs and are two examples.
We can represent all the solutions to by graphing the equation on a coordinate plane.
The graph is a line. All the points on the line are solutions to .
The inequality is an inequality in two variables. Its solution is any pair of and whose sum is 7 or less than 7.
This means it includes all the pairs that are solutions to the equation , but also many other pairs of and that add up to a value less than 7. The pairs and are two examples.
On a coordinate plane, the solution to includes the line that represents . If we plot a few other pairs that make the inequality true, such as and , we see that these points fall on one side of the line. (In contrast, pairs that make the inequality false fall on the other side of the line.)
We can shade that region on one side of the line to indicate that all points in it are solutions.
Inequality graphed on a coordinate plane, origin O. Each axis from negative 8 to 10, by 2’s. Solid line goes through 0 comma 7 and 5 comma 2. The region below the solid line is shaded. The points negative 6 comma 0 and 4 comma negative 7 are labeled.
What about the inequality ?
The solution is any pair of and whose sum is less than 7. This means pairs like and are not solutions.
On a coordinate plane, the solution does not include points on the line that represent (because those points are and pairs whose sum is 7).
To exclude points on that boundary line, we can use a dashed line.
All points below that line are pairs that make true. The region on that side of the line can be shaded to show that it contains the solutions.
Inequality graphed on a coordinate plane, origin O. Each axis from negative 8 to 10, by 2’s. Dashed line goes through 0 comma 7 and 7 comma 0. The region below the dashed line is shaded.