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3.07 Systems of Equations (2x2) and Inequalities
College Algebra: One Step at a Time, Pages 430-437: #13, 16, 19, 20, 25, Extra Problem Dr. Robert J. Rapalje Seminole State College of Florida Sanford, FL 32773
In each problem involving inequalities, there are three steps. First, you must get the line in place, by graphing the corresponding equation. Second, you must decide whether the line should be included or not--that is, should it be a dotted or solid line. Third, you must decide whether to shade above or below the line. In later problems, there is a fourth step, in which, when there are two or more inequalities, you must decide whether to shade the Union or the Intersection of the areas.
NOTE: You MUST have a positive Y coefficient!! If you have a negative coefficient, this REVERSES the RULE!! Unfortunately, in the format of this website, I have not learned how to make a dotted line. I will have to ask YOU to make the lines dotted that have either a < or > symbol.
Example.
Solution:
Step 1: Graph the line:
Since this is in standard form, find the x and y intercepts. If x = 0, then 2y = -12 y = -6 If y = 0, then 3x = -12 x = -4 Step 2: Graph this line with a dotted line (you will have to do this for me, since I don’t know how to graph a dotted line in this format.
(Dotted Line!) Step 3: Shade above the line. Don’t forget to make this line dotted!!
Example.
Solution:
Step 1: Graph the
line:
Since this is in standard form, find the x and y intercepts. If x = 0, then y = 8 If y = 0, then -2x = 8 x = -4 Step 2: Graph this line with a solid line.
(Solid Line!) Step 3: Shade below the line. Don’t forget to make this line solid!!
p. 433
#13.
Solution: Preliminary Step: Multiply both sides of the second equation by -1:
Step 1: Graph the first
line:
Since the first line is in standard form, find the x and y-intercepts. If x = 0, then -3y = -12 y = 4 If y = 0, then 2x = -12 x = -6 Step 2: Graph this line with a solid line.
(Solid Line!) Step 3: Since this is a negative y coefficient, the rule is reversed. Shade below the line! Don’t forget to make this line solid!!
Now, do the same process with the second inequality (use the same xy axes as the first):
Step 1: Graph the
second line:
Since the second line is in slope-intercept form, find the y-intercept and the slope. The y intercept is 8, and the slope is -4.
Step 2: Graph this line with a solid line.
(Solid Line!) Step 3: Shade above this line! Don’t forget to make this line solid!! Step 4: Shade the UNION of the two regions. This includes EVERYTHING that was shaded in either of the two graphs. It should look like this:
Final Answer: You must shade everything that is shaded on this graph.
p. 433 #16.
Solution:
Step 1: Graph the first
line:
Since the first line is in standard form, find the x and y-intercepts. If x = 0, then -y = -8 y = 8 If y = 0, then 4x = -8 x = -2 Step 2: Graph this line with a dotted line.
(Dotted Line!)
Step 3: Since this is a negative y-coefficient, the rule is reversed. Shade below the line! Don’t forget to make this line dotted!!
Now, do the same process with the second inequality (use the same xy axes as the first):
Step 1:
Graph the second line:
Since the second line is in standard form, find the x and y-intercepts. If x = 0, then 2y = -4 y = -2 If y = 0, then x = -4 Step 2: Graph this line with a dotted line.
(Dotted Line!)
Step 3: Shade below this line! Don’t forget to make this line dotted!! Step 4: Shade the INTERSECTION of the two regions. This includes ONLY the region common to both, the cross-shaded area ONLY. It should look like this:
Final Answer : You must shade ONLY the cross-shaded region on this graph! (That would be the lower right corner of the graph ONLY! Also, don’t forget to draw dotted lines!!)
p. 434 #19.
Solution:
Step 1: Graph
the first line: Since the first line is in standard form, find the x and y-intercepts. If x = 0, then -2y = 12 y = -6 If y = 0, then 3x = 12 x = 4 Step 2: Graph this line with a solid line.
(Solid Line!)
Step 3: Since this is a negative y-coefficient, the rule is reversed. Shade above the line! Don’t forget to make this line solid!!
Now, do the same process with the second and third inequality (use the same xy axes as the first):
Graph
the second line: This is a vertical line, the y-axis, make it a solid line, and shade to the right.
Graph the
third line:
This is a horizontal line, the x-axis, make it a solid line, and shade below the line.
(Solid Line) (Solid Line!)
Step 4: Shade the INTERSECTION of the three regions. This includes ONLY the region common to all three, the triple-shaded area ONLY. It should look like this:
Final Answer: You must shade ONLY the triple-shaded region on this graph! (That would be the triangular shaped region in the graph ONLY!) Also, don’t forget to draw solid lines!!
p.
434 #20.
Solution:
Step 1: Graph
the first line: Since the first line is in standard form, find the x and y-intercepts. If x = 0, then 3y = -6 y = -2 If y = 0, then x = -6
Step 2: Graph this line with a dotted line.
(Dotted Line!)
Step 3: This is a positive y-coefficient, with a “>” symbol. Shade above the line! Don’t forget to make this line dotted!!
Now, do the same process with the second and third inequality (use the same xy axes as the first):
Graph
the second line: This is a vertical line, 2 units to the right of the y-axis, make it a dotted line, and shade to the left.
Graph the
third line:
This is a horizontal line, up 3 units above the x-axis, make it a dotted line, and shade below the line.
(Dotted Line) (Dotted Line!)
Step 4: Shade the INTERSECTION of the three regions. This includes ONLY the region common to all three, the triple-shaded area ONLY. It should look like this:
Final Answer: You must shade ONLY the triple-shaded region on this graph! (That would be the triangular shaped region in the graph ONLY!) Also, don’t forget to draw dotted lines!!
p. 436 #25.
Solution:
Step 1:
Graph the first line:
Since the first line is in standard form, find the x and y-intercepts. If x = 0, then 2y = -4 y = -2 If y = 0, then x = -4
Step 2: Graph this line with a dotted line.
(Dotted Line!)
Step 3: This is a positive y-coefficient, with “<”. Shade below the line! Don’t forget to make this line dotted!!
Now, do the same process with the second inequality (use the same xy axes as the first):
Step 1: Graph
the second line: Since the second line is in standard form, find the x and y-intercepts. If x = 0, then -2y = -4 y = 2 If y = 0, then x = -4
Step 2: Graph this line with a dotted line.
(Dotted Line!)
Step 3: This is a negative y-coefficient, with “>”, so the rule is REVERSED! Shade below the line! Don’t forget to 0 this line dotted!!
Now, do the same process with the third inequality (use the same xy axes as the first two):
Step 1: Graph
the third line:
Since the first line is in slope intercept form, find the slope and y intercept. The y-intercept is 2 and the slope is -3. Step 2: Graph this line with a solid line.
(Dotted Line!)
Step 3: Shade below the line! Don’t forget to make this line solid!!
Step 4: Shade the INTERSECTION of the three regions. This includes ONLY the region common to all three, the triple-shaded area ONLY. It should look like this:
Final Answer: You must shade ONLY the triple-shaded region on this graph! (That would be the infinite region in the lower left part of the graph ONLY!) Also, don’t forget to draw dotted lines!!
EXTRA PROBLEM: Find the INTERSECTION of the regions.
Solution:
Step 1: Graph
the first line:
Since the first line is in standard form, find the x and y-intercepts. If x = 0, then -y = 2 y = -2 If y = 0, then 3x = 2 x = 2/3
Step 2: Graph this line with a dotted line.
(Dotted Line!)
Step 3: Since this is a negative y-coefficient, the rule is reversed. Shade above the line! Don’t forget to make this line dotted!!
Now, do the same process with the second inequality (use the same xy axes as the first):
Step 1: Graph
the second line: Since the second line is in standard form, find the x and y-intercepts. If x = 0, then y = 2 If y = 0, then x = 2
Step 2: Graph this line with a dotted line.
(Dotted Line!)
Step 3: Shade above this line! Don’t forget to make this line dotted!!
Step 4: Shade the INTERSECTION of the two regions. This includes ONLY the region common to both, the cross-shaded area ONLY. It should look like this:
Final Answer: You must shade ONLY the cross-shaded region on this graph! (That would be the triangular-shaped region in the upper middle of the graph!) Also, don’t forget to draw dotted lines!!
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