Example #2
Sometimes a system can be used to represent inequalities. They are graphed just like equations but must be shaded to represent the area of solutions. Look at this system of inequalities:
x + y > 8
2x + y ≤ 12
The steps given next are for graphing by hand to help with the next lesson. However, this is easily done in Geogebra and on a graphing calculator.
Solve the following system of inequalities by graphing.
x + y > 8
Steps for graphing:
1. Solve for y in each equation.
2. Graph the two inequalities as if they were equations. Remember the first should be a dotted line and the second should be a solid because of the inequality symbols
3. Pick a point on either side of each line to determine where to shade. Plug the point into the original inequality. If it is a true statement, shade over the point. If it is a false statement, shade opposite to the point.
4. The solution is the overlapping shaded area.
OK, now let’s do this step by step.
Step 1: Solving for y in each equation, we would have
y > _ - _
y ≤ _ - _
Answer and explanation:
y > 8 – x
x + y > 8 - substract x from each side to isolate y
x - x + y > 8 - x - combine like terms
y > 8 - x
y ≤ 12 – 2x
2x + y ≤ 12 - subtract 2x from each side to isolate y
2x - 2x + y ≤ 12 - 2x -combine like terms
y ≤ 12 - 2x
Step 2: Graph the two inequalities as if they were equations. Remember the first should be a dotted line and the second should be a solid because of the inequality symbols
Answer:
Step 3: Pick a point on either side of each line to determine where to shade. Plug the point into the original inequality. If it is a true statement, shade over the point. If it is a false statement, shade opposite to the point.
Let’s do y > 8 – x first.
Answer:
Plugging (0, 0) into this inequality, we have 0 > 8 - 0 and this is false. Therefore, we shade the side of the line that does not contain (0,0).
Do the same for the second equation.
y ≤ 12 – 2x
Answer:
Plugging (0,0) into this inequality, we have
0 ≤ 12 – 0 and this is true. Therefore, we shade over the point (0,0).
Step 4: The solution is the overlapping shaded area.
View solution here.
