L1.3 B1 Solving Rational Equations - Part 1
Completion requirements
Unit 1
Functions
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Part 1.3B corresponds to section 9.3, starting on page 457 of your Pre-Calculus 12 textbook.
Many situations can be modelled by rational functions. An oil spill from an ocean tanker or the speeds associated with a highway accident are examples of situations that can be modelled by rational functions.
The corresponding rational equations can be solved algebraically and graphically.
In earlier lessons, you learned how to solve radical and polynomial equations. Solving rational equations is very similar. Restrictions on the variable should be determined, and then the equation can be solved.
The corresponding rational equations can be solved algebraically and graphically.
In earlier lessons, you learned how to solve radical and polynomial equations. Solving rational equations is very similar. Restrictions on the variable should be determined, and then the equation can be solved.
Solve «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»=«/mo»«mfrac»«mn»14«/mn»«mi»x«/mi»«/mfrac»«mo»§#8722;«/mo»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math» algebraically.
Determine any restrictions on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math».
Multiply every term by the lowest common denominator so the denominators are eliminated from the equation.
Verify «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math».
The left side equals the right side when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math» is used in the equation. Therefore, it is a solution.
Verify «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mn»2«/mn»«/mstyle»«/math».
The left side equals the right side when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mn»2«/mn»«/mstyle»«/math» is used in the equation. Therefore, it is a solution.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math»
Multiply every term by the lowest common denominator so the denominators are eliminated from the equation.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mn»0«/mn»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»14«/mn»«menclose notation=¨updiagonalstrike¨»«mfenced»«mi»x«/mi»«/mfenced»«/menclose»«/mfrac»«menclose notation=¨updiagonalstrike¨»«mfenced»«mi»x«/mi»«/mfenced»«/menclose»«mo»§#8722;«/mo»«mi»x«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»+«/mo»«mn»5«/mn»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»14«/mn»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»5«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»14«/mn»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»5«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»14«/mn»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»x«/mi»«mo»§#8722;«/mo»«mn»7«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»-«/mo»«mn»7«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»7«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Neither of the possible solutions are non-permissible values, so it is reasonable to think they are both solutions to the given equation. Each value can be verified using substitution.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»-«/mo»«mn»7«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»7«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Verify «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math».
| Left Side | Right Side |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨left¨»«mtr»«mtd»«mfrac»«mn»14«/mn»«mi»x«/mi»«/mfrac»«mo»§#8722;«/mo»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mn»14«/mn»«mfenced»«mn»7«/mn»«/mfenced»«/mfrac»«mo»§#8722;«/mo»«mfenced»«mn»7«/mn»«/mfenced»«mo»+«/mo»«mn»5«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»2«/mn»«mo»§#8722;«/mo»«mn»7«/mn»«mo»+«/mo»«mn»5«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math» |
The left side equals the right side when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math» is used in the equation. Therefore, it is a solution.
Verify «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mn»2«/mn»«/mstyle»«/math».
| Left Side | Right Side |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨left¨»«mtr»«mtd»«mfrac»«mn»14«/mn»«mi»x«/mi»«/mfrac»«mo»§#8722;«/mo»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mn»14«/mn»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mfrac»«mo»§#8722;«/mo»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»+«/mo»«mn»5«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo»§#8722;«/mo»«mn»7«/mn»«mo»+«/mo»«mn»2«/mn»«mo»+«/mo»«mn»5«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math» |
The left side equals the right side when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mn»2«/mn»«/mstyle»«/math» is used in the equation. Therefore, it is a solution.