Transcript document

More Linear
Equations
Solving Rational Equations
Example 1
3( x  4)  5( x  6)  2( x  1)
3x  12  5x  30  2x  2
3x  12  7 x  32
 3x
 3x
12  4x  32
 32
 32
44  4x
11  x
Before working this
problem, get rid of
the parentheses by
distributing the
outside numbers.
If there are like terms
on the same side of
the equals sign, just
combine them as
they are.
Solving Rational Equations
Example 2
2( x  1)  3( x  2)  4( x  3)
2x  2  3x  6  4x 12
2x  2  7 x  6
 2x
 2x
2  5x  6
6
6
8  5x
8
x
5
Before working this
problem, get rid of
the parentheses by
distributing the
outside numbers.
If there are like terms
on the same side of
the equals sign, just
combine them as
they are.
Solving Rational Equations
Example 3
4( x  2)  5( x  3)  2( x  1)
4x  8  5x  15  2x  2
4x  8  3x  17
 3x
 3x
x  8  17
8 8
x  25
Before working this
problem, get rid of
the parentheses by
distributing the
outside numbers.
If there are like terms
on the same side of
the equals sign, just
combine them as
they are.
Solving Rational Equations
Example 4
3( x  1)  2( x  5)  5( x  1)  2( x  3)
3x  3  2x  10  5x  5  2x  6
5x  7  3x 11
 3x
 3x
2x  7  11
7 7
2x  4
x2
Don’t forget to
multiply by the
number and the sign
in front of the
number.
Solving Rational Equations
Example 5
x  3( x  5)  4( x  3)  ( x  4) Before working this
problem, get rid of
x  3x 15  4x 12  x  4 the parentheses by
 2x 15  3x  8
 2x
 2x
15  5x  8
8
8
23  5x
23
x
5
distributing the
outside numbers.
If there are like terms
on the same side of
the equals sign, just
combine them as
they are.
Solving Rational Equations
Example 6
x 3 x 1
  
3 4 4 2
3
8   6
4
3
20   12
5
4
9  
9
4
5
12  
6
1
8  
4
Fractions!!!
10
2
 2  14
7  
 11  11
You only need to know one
“trick” to solve equations
with fractions. To multiply a
fraction times a whole
number, divide the bottom
into the whole number, then
multiply what’s left.
Let’s practice a couple of
those before we work this
equation.
Solving Rational Equations
Example 6
x 3 x 1
12 (    )
3 4 4 2
4x  9  3x  6
 3x
 3x
x  9  6
9 9
x  15
Back to the problem.
Find the common denominator of
all of the fractions and multiply
everything by that.
Remember, to multiply fractions
times whole numbers, divide the
bottom into the whole number,
then multiply what’s left.
Solving Rational Equations
Example 7
x 3 x 3
10 (    )
2 5 5 2
5x  6  2x  15
 2x  2x
3x  6  15
6 6
3x  9
x  3
Find the common denominator of
all of the fractions and multiply
everything by that.
Remember, to multiply fractions
times whole numbers, divide the
bottom into the whole number,
then multiply what’s left.
Solving Rational Equations
Example 8
x 3
x
15 (   2  )
3 5
5
5x  9  30  3x
 3x
 3x
8x  9  30
9 9
8x  21
21
x
8
Remember, to multiply fractions
times whole numbers, divide the
bottom into the whole number,
then multiply what’s left.
Solving Rational Equations
Example 10
3( x  4)  3( x  6)
3x  12  3x 18
 3x
 3x
12  18
O
What happened that
is different from the
other problems?
If the variables go away, then
the answer is either “no
solutions” or “all real numbers”.
It’s “no solutions” if what’s
leftover is a false statement.
It’s “all real numbers” if what’s
leftover is a true statement.
Solving Rational Equations
Example 11
2(3x  6)  3(2 x  4)
6x  12  6x  12
 6x
 6x
12  12
ALL REAL NUMBERS
Remember, if the
variables go away, then
the answer is either “no
solutions” or “all real
numbers”.
It’s “no solutions” if
what’s leftover is a false
statement.
It’s “all real numbers” if
what’s leftover is a true
statement.
Solving Rational Equations
Example 12
5( x  2)  2( x  5)
5x  10  2x  10
 2x
 2x
3x  10  10
 10  10
3x  0
3 3
x0
Is this “no solutions” or
“all real numbers”?
Neither. Remember, it’s
only one of those if the
VARIABLES go away.
Just keep on working
this problem.
Solving Rational Equations
Pair Practice: Solve these problems
with a partner.
1)
2)
3)
4)
5)
5x 10  25
3x  4  5x  12
Answers :
1)3
2) 8
3)7
9
4)
8
5)
17
6)
2
4( x  3)  2( x  1)
3( x  2)  2( x  4)  x  3  4( x  1)
5(2 x  3)  2(5 x  1)
x 3 x 2
6) 3  4  2  3
Solving Rational Equations
Answers to worksheet #1: 1) 3
2)  36
3)  17
4)  12
8
5)
5
6)  4
13
7)
4
3
8)
5
9)3
10)
 14
11)
3
12)
13)
32
14)
3
1
15)
4
 234
16)
25
The Rules are very simple. Be the first person to identify
the mistake made in the following problems, raise your
hand and hollar “Hey, Stupid!” The first correct answer
wins you five bonus points on the Chapter Test. Once you
win once, you cannot win again until next chapter.
It’s time for…
HEY STUPID!!!
Solving Rational Equations
HEY STUPID!!!
#1
5( x  1)  3( x  3)
5x  5  3x  3
 3x
 3x
2x  5  3
5 5
2x  8
2 2
x  4
Didn’t distribute the 3 in
the first step.
Solving Rational Equations
HEY STUPID!!!
#2
2x  5  3
3 3
2x  2
2 2
x 1
Supposed to subtract
the 5 instead of the 3.
Solving Rational Equations
HEY STUPID!!!
#3
5(3x  1)  3(5 x  3)
15x  5  15x  9
 15x
 15x
5  9
ALL REAL NUMBERS
Should be no solutions,
because the statement
that is leftover is NOT
true.
Solving Rational Equations
HEY STUPID!!!
#4
4( x  1)  3( x  3)  2( x  1)
4x  4  3x  9  2x  2
4x  4  x 11
Didn’t distribute the
x
x
negative in the first
step.
3x  4  11
4 4
3x  15
3 3
x  5
Solving Inequalities
Hey, Stupid! #5
9  3x  21
9
9
 3x  12
3 3
x  4
5  4 3
Anytime you multiply or divide by
a negative, you have to turn the
inequality symbol around.
Solving Rational Equations
Hey, Stupid!
#6
x 3
x
6 (   4 )
3 2
2
2 x  9  4  3x
 3x
 3x
5x  9  4
9 9
5x  5
x  1
You have to multiply
EVERYTHING by the common
denominator. They didn’t multiply
the four by six.
Solving Inequalities
Hey, Stupid! #7
3x  2  4x  8
 3x
 3x
 2  x 8
8
8
10  x
9 10 11
When you have the x on the left,
you need to rewrite the WHOLE
problem by turning it all around
before graphing it.
You should be graphing :
x  10