II. GRAPHING. Graph the given inequality below, then shade the region of points that satisfies the inequality Use the provided Cartesian plane. (6 pts.) 9. уѕх - 4 PROBLEM SOLVING. Solve for the following function notations. Show your solution.
To graph a linear inequality in two variables (say, x and y ), first get y alone on one side. Then consider the related equation obtained by changing the inequality sign to an equality sign. The graph of this equation is a line.
If the inequality is strict ( < or > ), graph a dashed line. If the inequality is not strict ( ≤ or ≥ ), graph a solid line.
Finally, pick one point that is not on either line ( (0,0) is usually the easiest) and decide whether these coordinates satisfy the inequality or not. If they do, shade the half-plane containing that point. If they don't, shade the other half-plane.
Graph each of the inequalities in the system in a similar way. The solution of the system of inequalities is the intersection region of all the solutions in the system.
Example 1:
Solve the system of inequalities by graphing:
y≤x−2y>−3x+5
First, graph the inequality y≤x−2 . The related equation is y=x−2 .
Since the inequality is ≤ , not a strict one, the border line is solid.
Graph the straight line.
Consider a point that is not on the line - say, (0,0) - and substitute in the inequality y≤x−2 .
0≤0−20≤−2
This is false. So, the solution does not contain the point (0,0) . Shade the lower half of the line.
Similarly, draw a dashed line for the related equation of the second inequality y>−3x+5 which has a strict inequality. The point (0,0) does not satisfy the inequality, so shade the half that does not contain the point (0,0) .
The solution of the system of inequalities is the intersection region of the solutions of the two inequalities.
Answers & Comments
Answer:
Graphing Systems of Linear Inequalities
To graph a linear inequality in two variables (say, x and y ), first get y alone on one side. Then consider the related equation obtained by changing the inequality sign to an equality sign. The graph of this equation is a line.
If the inequality is strict ( < or > ), graph a dashed line. If the inequality is not strict ( ≤ or ≥ ), graph a solid line.
Finally, pick one point that is not on either line ( (0,0) is usually the easiest) and decide whether these coordinates satisfy the inequality or not. If they do, shade the half-plane containing that point. If they don't, shade the other half-plane.
Graph each of the inequalities in the system in a similar way. The solution of the system of inequalities is the intersection region of all the solutions in the system.
Example 1:
Solve the system of inequalities by graphing:
y≤x−2y>−3x+5
First, graph the inequality y≤x−2 . The related equation is y=x−2 .
Since the inequality is ≤ , not a strict one, the border line is solid.
Graph the straight line.
Consider a point that is not on the line - say, (0,0) - and substitute in the inequality y≤x−2 .
0≤0−20≤−2
This is false. So, the solution does not contain the point (0,0) . Shade the lower half of the line.
Similarly, draw a dashed line for the related equation of the second inequality y>−3x+5 which has a strict inequality. The point (0,0) does not satisfy the inequality, so shade the half that does not contain the point (0,0) .
The solution of the system of inequalities is the intersection region of the solutions of the two inequalities.