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Warm-Up Exercises
Solve the equation.
1. 4 = 6
x 15
ANSWER
10
2. 5(3x – 2) = 4(4x – 1)
ANSWER
–6
Warm-Up Exercises
Solve the equation.
3. x(x – 4) = 3(x – 4) + x
ANSWER
2,6
4. 2x2 + 5x – 12 = 0
ANSWER
– 4,
3
2
Warm-Up Exercises
Solve the equation.
5. How much peroxide is there in 30 liters of a 45%
peroxide solution?
ANSWER
13.5 L
Warm-Up1Exercises
EXAMPLE
Solve a rational equation by cross multiplying
Solve:
3 = 9
x + 1 4x + 1
3 = 9
x + 1 4x + 1
Write original equation.
3(4x + 1) = 9(x + 1)
Cross multiply.
12x + 3 = 9x + 9
3x + 3 = 9
3x = 6
x=2
Distributive property
Subtract 9x from each side.
Subtract 15 from each side.
Divide each side by 3.
ANSWER
The solution is 2. Check this in the original equation.
Warm-Up
YOU
TRY Exercises
for Examples 1
1. 3 = 2
5x x – 7
SOLUTION
3
2
5x = x – 7
3(x – 7) = 2(5x )
3x – 21 = 10x
– 7x – 21 = 0
– 7x = 21
x=–3
Write original equation.
Cross multiply.
Distributive property
Subtract 10x from each side.
Subtract 21 from each side.
Divide each side by 7.
ANSWER
The solution is – 3 . Check this in the original equation
Warm-Up
YOU
TRY Exercises
for Examples 1
2. – 4 = 5
x–3
x+3
SOLUTION
–4 = 5
x–3
x+3
– 4(x – 3) = 5(x + 3)
Write original equation.
Cross multiply.
Distributive property
– 4x + 12 = 5x + 15
– 9x + 12 = 15
Subtract 5x from each side.
– 9x = 3
Subtract 12 from each side.
Divide each side by 9.
x= –1
3
ANSWER
The solution is – 1 . Check this in the original equation
3
Warm-Up
YOU
TRY Exercises
for Examples 1
3. 1 = x
2x +5 11x+ 8
SOLUTION
1 = x
2x +5 11x+ 8
Write original equation.
1(11x + 8) = x(2x + 5)
Cross multiply.
11x + 8 = 2x2 + 5x
Distributive property
2x2 – 6x2 +8 = 0
x2 – 3x2 +4= 0
(x – 4)(x + 1) = 0
Subtract 2x2 , and 5 from each side.
Warm-Up
YOU
TRY Exercises
for Examples 1
x = 4 , x = –1
ANSWER
x = 4 , x = –1
Warm-Up2Exercises
EXAMPLE
Standardized Test Practice
SOLUTION
5+ 7 =– 9
x
x
4
4x( 5 + 7 ) = 4x – 9
x
x
4
20 + 7x = –36
7x = – 56
x=–8
Write original equation.
Multiply each side by the LCD, 4x.
Simplify.
Subtract 20 from each side.
Divide each side by 7.
Warm-Up2Exercises
EXAMPLE
Standardized Test Practice
ANSWER
The correct answer is B.
Warm-Up3Exercises
EXAMPLE
Solve a rational equation with two solutions
Solve: 1 –
8
3
=
x–5 x
8
3
Write original equation.
=
x–5 x
x(x – 5) 1– 8
x(x – 5) 3 Multiply each side by the LCD, x(x–5).
=
x
x–5
Simplify.
x(x –5) – 8x = 3(x – 5)
1–
(
)
x2 – 5x – 8x = 3x – 15
x2 – 16x +15 = 0
(x – 1)(x – 15) = 0
x = 1 or x = 15
Simplify.
Write in standard form.
Factor.
Zero product property
Warm-Up3Exercises
EXAMPLE
Solve a rational equation with two solutions
ANSWER
The solutions are 1 and 15. Check these in the original
equation.
Warm-Up4Exercises
EXAMPLE
Check for extraneous solutions
2
6
8x
– 4x
Solve: x – 3 = 2
x+3
x –9
SOLUTION
Write each denominator in factored form.
The LCD is (x + 3)(x – 3).
8x2
6 =
– 4x
x –3 (x + 3)(x – 3)
x+3
2
8x
6
(x + 3)(x – 3)
(x + 3)(x – 3)
(x + 3)(x – 3) 4x
=
x+3
x –3
(x + 3)(x – 3)
6(x + 3) = 8x2 – 4x(x – 3)
6x + 18 = 8x2 – 4x2 + 12x
Warm-Up4Exercises
EXAMPLE
Check for extraneous solutions
0 = 4x2 + 6x –18
0 = 2x2 + 3x – 9
0 = (2x – 3)(x + 3)
2x – 3 = 0 or x + 3 = 0
x = 3 or x = –3
2
You can use algebra or a graph to check whether
either of the two solutions is extraneous.
Algebra
The solution checks, 32 but the apparent solution –3 is
extraneous, because substituting it in the equation
results in division by zero, which is undefined.
Warm-Up4Exercises
EXAMPLE
Check for extraneous solutions
Graph
Graph y = x –6 3 and
6 = 8(– 3)2 – 4(–3)
–3 –3 (–3)2 – 9 –3 +3
2
8x
– 4x
y= 2
x –9
x+3
3
The graphs intersect when x = 2
but not when x = –3.
ANSWER
The solution is 23 .
Warm-Up
YOU
TRY Exercises
for Examples 2, 3 and 4
Solve the equation by using the LCD. Check for
extraneous solutions.
4. 7 + 3 = 3
x
2
SOLUTION
Write each denominator in factored form. The LCD is 2x
7 + 3 =3
x
2
2x
7 + 2x
2
3 = 2x 3
x
7x + 6 = 6x
x= –6
Warm-Up
YOU
TRY Exercises
for Examples 2, 3 and 4
5. 2 + 4 = 2
3
x
SOLUTION
Write each denominator in factored form. The LCD is 3x
2 + 4 =2
3
x
3x 2 + 3x 4
x
3
= 3x 2
6 + 4x = 6x
6 = 2x
x=3
Warm-Up
YOU
TRY Exercises
for Examples 2, 3 and 4
6. 3 + 8 = 1
x
7
SOLUTION
Write each denominator in factored form. The LCD is 7x
3 + 8 =1
x
7
7x
3 + 7x
7
8
x
= 7x 1
3x + 56 = 7x
56 = 4x
x = 14
Warm-Up
YOU
TRY Exercises
for Examples 2, 3 and 4
7. 3 + 4 = x +1
x –1 x –1
2
SOLUTION
Write each denominator in factored form. The LCD is
2( x – 1)
3 + 4 = x +1
x –1 x –1
2
(x – 1 )(2)
3
4 = (x – 1)(2) x + 1
+
(x
–
1)(2)
2
x –1
x1
3x – 3 + 8= 2x + 2
x=–3
Warm-Up
YOU
TRY Exercises
for Examples 2, 3 and 4
8. 3x – 5 = 3
x +1 2x
2x
SOLUTION
Write each denominator in factored form. The LCD is
(x + 1)(2x)
3x – 5 = 3
x +1 2x
2x
2x (x + 1)
3x – 2x (x +1) 5 = 2x (x +1) 3
2x
x +1
2x
Warm-Up
YOU
TRY Exercises
for Examples 2, 3 and 4
6x2 – 5x – 5 = 3x + 3
0 = 3x + 3 – 6x2 +5x + 5
0 = –6x2 - 8x - 8
0 = (3x +2) (x – 2)
3x + 2 = 0
x= – 2
3
or
x–2=0
x=2
Warm-Up
YOU
TRY Exercises
9.
for Examples 2, 3 and 4
5x = 7 + 10
x –2
x –2
SOLUTION
Write each denominator in factored form. The LCD is x – 2
5x = 7 + 10
x –2
x –2
10
x – 2 x5x–2 = (x – 2) 7 + (x – 2) x – 2
5x = 7x – 14 + 10
4 = 2x
x=2
x=2 results in no solution.
Warm-Up5Exercises
EXAMPLE
Solve a rational equation given a function
Video Game Sales
From 1995 through 2003, the annual sales S (in billions of
dollars) of entertainment software can be modeled by
2 + 3220
848t
S(t) = 115t2 + 1000 0 < t < 8
where t is the number of years since 1995. For which
year were the total sales of entertainment software
about $5.3 billion?
SOLUTION
2 + 3220
848t
S(t) = 115t2 + 1000 Write given function.
2 + 3220
848t
5.3 = 115t2 + 1000 Substitute 5.3 for S(t).
Warm-Up5Exercises
EXAMPLE
Solve a rational equation given a function
5.3(115t2 + 1000) = 848t2 + 3220
609.5t2 + 5300 = 848t2 + 3220
Multiply each side by
115t2 + 1000.
Simplify.
5300 = 238.5t2 + 3220
Subtract 609.5t2 from
each side.
2080 = 238.5t2
Subtract 3220 from
each side.
Divide each side
by 238.5.
8.72
t2
±2.95 t
Take square roots of
each side.
Because –2.95 is not in the domain (0 < t < 8),
the only solution is 2.95.
Warm-Up5Exercises
EXAMPLE
Solve a rational equation given a function
ANSWER
So, the total sales of entertainment software were
about $5.3 billion about 3 years after 1995, or in 1998.
Warm-Up
YOU
TRY Exercises
for Example 5
10. What If? Use the information in Example 6 to
determine in which year the total sales of
entertainment software were about $4.5 billion.
SOLUTION
2 + 3220
848t
S(t) =
115t2 + 1000
2 + 3220
848t
4.5 = 115t2 + 1000
4.5(115t2 + 1000) = 848t2 + 3220
517.5t2 + 4500 = 848t2 + 3220
1280 = 330.5t2
Write given function.
Substitute 5.3 for S(t).
Multiply each side by
115t2 + 1000.
Simplify.
Subtract.
Warm-Up
YOU
TRY Exercises
3.87 = + 2
±1.97
t
for Example 5
Divide each side
by 330.5.
Take square roots of
each side.
Because – 1.97 is not the domain (0 < t < 8), the only
solution is 1.97
So , the total sales of entertainment software were
about $ 4.5 million about two your after 1995, or in
1997
Warm-Up
Exercises
KEEP
GOING
Solve the equation.
6
x–2
ANSWER
4
6
12
=
x+3
x
ANSWER
– 12 , 3
5
1.
5
=
2x + 3
2.
5–
3.
1
2
3
+
=
x
x2
x3
ANSWER
–3,1
4.
x
x–2
38
+
=
x+1
x–1
x2 – 1
ANSWER
–4,5
Warm-Up
Exercises
KEEP
GOING
5.
So far this season, a baseball team has won 36 of
its first 64 games. How many consecutive games
does the need to win to raise its winning
percentage to 60%?
ANSWER
6 games