L1.3 B1 Solving Rational Equations - Part 4
Completion requirements
Unit 1
Functions
Solving Rational Equations Graphically
Like many other types of equations, rational equations can be solved graphically. One method is to graph functions that correspond to each side of the equation and then determine the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-value of any intersections. A second method is to rearrange the equation so one side is equal to zero, and then determine any «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts.
Solve «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mn»6«/mn»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math» by graphing.
Start by rearranging the equation so one side is equal to zero.
Next, graph «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mn»6«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math», and determine the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts.
Clearly, the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts are not integers.
Use your calculator or other graphing technology to determine the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts.
One «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept is at approximately «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«mo».«/mo»«mn»63«/mn»«/mstyle»«/math».
Another «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept is at approximately «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»7«/mn»«mo».«/mo»«mn»37«/mn»«/mstyle»«/math».
There is another method of solving an equation graphically.
The equation given was «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mn»6«/mn»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math».
Instead of rearranging the equation so one side is equal to zero, and then graphing the corresponding function, treat the left and right sides of the equation as separate functions to be graphed. The «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-coordinates of the points of intersection of the two graphs represent the solutions to the equation.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math»
Therefore, graph «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math», and then determine where the two graphs intersect.
These two functions intersect at the same «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-values as the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts of the graph of the single function used in the first method.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mn»6«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math»
Next, graph «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mn»6«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math», and determine the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts.
Clearly, the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts are not integers.
Use your calculator or other graphing technology to determine the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts.
One «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept is at approximately «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«mo».«/mo»«mn»63«/mn»«/mstyle»«/math».
Another «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept is at approximately «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»7«/mn»«mo».«/mo»«mn»37«/mn»«/mstyle»«/math».
There is another method of solving an equation graphically.
The equation given was «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mn»6«/mn»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math».
Instead of rearranging the equation so one side is equal to zero, and then graphing the corresponding function, treat the left and right sides of the equation as separate functions to be graphed. The «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-coordinates of the points of intersection of the two graphs represent the solutions to the equation.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math»
Therefore, graph «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mfrac»«mi»x«/mi»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mn»6«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math», and then determine where the two graphs intersect.
These two functions intersect at the same «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-values as the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercepts of the graph of the single function used in the first method.