Download - 5.6 Solve Rational Equations
5.6 Solve Rational Equations5.6 Solve Rational Equationsp. 345p. 345
What is the easiest way to solve a rational What is the easiest way to solve a rational equation?equation?
What is an extraneous solution?What is an extraneous solution?
How do you know when to cross multiply or How do you know when to cross multiply or multiply out the denominator?multiply out the denominator?
Cross Multiplying
When there is only one fraction on each side of the =, When there is only one fraction on each side of the =, just cross multiply as if you are solving a proportion.just cross multiply as if you are solving a proportion.
Beware of extraneous solutions!Beware of extraneous solutions!
Solve: 3x + 1=
94x + 1
3x + 1=
94x + 1 Write original equation.
3(4x + 5) = 9(x + 1) Cross multiply.12x + 15 = 9x + 9 Distributive property3x + 15 = 9 Subtract 9x from each side.
3x = – 6 Subtract 15 from each side.x = – 2 Divide each side by 3.
The solution is –2. Check this in the original equation.
ANSWER
Example: Solve.
4
1
4
32
xxx
xx 42 12342 xxx
0122 xx0)3)(4( xx
03or 04 xx3or 4 xx
Check your solutions!Check your solutions!
)4(3 x
Solve 1
2
22
62
x
x
xx
)1(6 x)2)(1(2 xxx
6)2(2 xx3)2( xx
0322 xx0)1)(3( xx
01or 03 xx1or 3 xx
1
2
)1(2
6
x
x
xx
Check your solutions!Check your solutions!
An alloy is formed by mixing two or more metals. Sterling silver is an alloy composed of 92.5% silver and 7.5% copper by weight. Jewelry silver is composed of 80% silver and 20% copper by weight. How much pure silver should you mix with 15 ounces of jewelry silver to make sterling silver?
ALLOYS
SOLUTION
Percent of copper in mixture =Weight of copper in mixture
Total weight of mixture
Page 345
7.5100
0.2(15)15 + x= x is the amount of silver added.
7.5(15 + x) = 100(0.2)(15) Cross multiply.
112.5 + 7.5x = 300 Simplify.7.5x = 187.5 Subtract 112.5 from each side.
x = 25 Divide each side by 7.5.
You should mix 25 ounces of pure silver with the jewelry silver.
ANSWER
Hint for solving:Hint for solving:
• A quick way to solve a rational A quick way to solve a rational equation is to multiply everything equation is to multiply everything through by the LCD. through by the LCD.
• This will get rid of all the fractions!This will get rid of all the fractions!
• Beware of extraneous solutions!Beware of extraneous solutions!
Example: Solve.
xx
12
2
13
LCD: 2x
Multiply each fraction through by the LCD
x
xx
x
x 12*2
2
23*2
246 x
18 x
18xCheck your solution!Check your solution!
18
12
2
1
18
3
1293
Solve: 1 – 8x – 5
=3x
1 – 8x – 5
=3x
Write original equation.
x(x – 5) 1– 8x – 5( ) = x(x – 5) 3
xMultiply each side by the LCD, x(x–5).
x(x –5) – 8x = 3(x – 5) Simplify.
x2 – 5x – 8x = 3x – 15 Simplify.
x2 – 16x +15 = 0 Write in standard form.
(x – 1)(x – 15) = 0Factor.
x = 1 or x = 15 Zero product property
The solutions are 1 and 15. Check these in the original equation.
ANSWER
Solve.1
54
1
5
xx
x LCD: ?LCD: (x+1)
)1(
)1(5)1(4
)1(
)1(5
x
xx
x
xx
5445 xx
145 xx
1x
Check your solution!Check your solution!
11
54
11
)1(5
0
54
0
5
?
No Solution!No Solution!
Solve. 14
6
2
232
xx
x
Factor 1st!
1)2)(2(
6
2
23
xxx
x
LCD: (x+2)(x-2)
)2)(2()2)(2(
)2)(2(6
)2(
)2)(2)(23(
xx
xx
xx
x
xxx
42264263 22 xxxxxx
2443 22 xxx
0642 2 xx0322 xx
0)1)(3( xx
01or 03 xx
1or 3 xx
Check your solutions!Check your solutions!
• What is the easiest way to solve a rational What is the easiest way to solve a rational equation?equation?
Multiply the equation by a value that will Multiply the equation by a value that will make the denominator equal to 1.make the denominator equal to 1.
• What is an extraneous solution?What is an extraneous solution?
A solution that does not make a true A solution that does not make a true statement in the original equation.statement in the original equation.
• How do you know when to cross multiply How do you know when to cross multiply or multiply out the denominator?or multiply out the denominator?
Cross multiply if there is only one fraction Cross multiply if there is only one fraction on each side of the equal sign, otherwise on each side of the equal sign, otherwise multiply out the denominator.multiply out the denominator.
5.6 Assignment
p. 348
5-27 odd