# 1078 Basic Algebra questions: solving an equation and equalities ```1078 Basic Algebra questions: solving an equation,
equalities and inequalities
Edward Morey October 6, 2006
(note that the questions are numbered in groups)
_________________________________________________________
Solve for the indicated variable in terms of the remaining variables:
1. A= P+Prt; for r
y  mx  b ; for m
1. Solve for x:
log b (2 x)  12 log b 16  log b 2
1
log b (2 x)  log b 16 2  log b 2
log b (2 x)  log b 4  log b 2
4
 log b 2
2
So 2 x  2  x  1
log b (2 x)  log b
1. Solve this equation for u: 2 
5 3

u u2
5 3

times u 2 on both sides  u 2  2    2 u 2
u u

2
 2u  5u   3
2
5 3

u u2
2u 2  5u  3  0
 u  3 2u  1  0   u  3  0
u  3 or u 
1
2
or
 2u  1  0
2. You need a compressor and there are two choices for you. You can rent a compressor
for the cost of \$90/day. Alternatively, you can buy one at \$4,000 with maintenance
costs around \$10/day. Let X denote the number of days that the compressor is needed.
Find out the value of X that makes these two choices equivalent.
Total cost for buying a compressor after x days: 4,000+10x
Total cost for renting one after x days:
90x
Set two costs function equalized and solve for x:
4000+10x = 90x  80x=4000  x=50 days
1. Solve for x.
(A) 3x  5  2
(B)
2
1
1
x x
3
2
4
(C) x 2  3x  0
1. Supply and demand
Suppose that the supply and demand equations for printed T-shirts in a little town during
Christmas week are
Supply: p = q + 12
Demand: p = - 2q + 36
Where p is the price in dollars and q is the quantity in hundreds.
A) Find the supply and demand quantity if p = \$16.
B) Find the equilibrium price and quantity. (when the price of supply equals the price
of demand).
1. Solve.
x 9
x


5 10 2
A x  3
Solve and graph
1.  8  2x  4  12
A 2 x 8
(
]
–2
8
x
2
1  3  y  9
3
A 9  y  6
(
]
–9
6
y
3. You need a compressor and there are two choices for you. You can rent a compressor
for the cost of \$80/day. Alternatively, you can buy one at \$7,000 with maintenance
costs around \$10/day. Let X denote the number of days that the compressor is needed.
Find out the value of X that makes these two choices equivalent.
A Solving 80x  10x  7000 , x = 100 days.
Check the following equalities.
2
1
x4 x4
3
2
y y 1
2.  
4 3 2
1.
3. 3(x-2 ) = 2(x-3) + x
4.
2
1
x 1  x  3
3
2
5.  1 
2
y  5  11
3
6. 10  8  3u  6
Demonstrate the equalities:
1.
a, a 2  2ab  b 2  (a  b) 2
b, a 2  2ab  b 2  (a  b) 2
c, a 2  b 2  (a  b)(a  b)
d , a 3  b3  (a  b)(a 2  ab  b 2 )
e, a 3  b3  (a  b)(a 2  ab  b 2 )
3
5  4  y  10
2
3
5  4   y  10  4
2
3
9   y  6
2
 2   2  3 
 2
9          y   6   
 3   3  2 
 3
6  y  4
or
3
3
y y9 y6
2
2
3
3
62
4  y  10   y  6  y  
 y  4
2
2
3
Combine : 4  y  6
5  4 
Solve for x:
1. x 2  20  5
2. 2 x 2  7 x  3  0
3. 2 x 2  3 x  x 2  2 x  12
4. 2 x 2  20 x  6  0
1. The supply for broccoli is described by the equation Q = P/6 where P is the price of broccoli
and Q is the amount that will be supplied at price P. The demand equation is described by Q
= 330-9P where P is the price of broccoli and Q is the amount that will be demanded. What is
the equilibrium PRICE of broccoli where demand equals supply?
2. Solve for y in terms of x the expression ( y  3x) 2  9 x 2
3. Solve by factoring:
x 2  2 x  35  0
A (Since ( x  7)( x  5)  0 ,) x  7,5
4. Solve using the quadratic formula
x 2  7 x  11  0
7 5
A
2
5. Suppose the demand equation (inverse demand function) is written as:
D  150 / p
And the supply equation is:
S  10 p  20
What is the equilibrium price p when D = S. (Choose only the positive price.)
A (By ( p  5)( p  3)  0 ,) p  5 .
1. Solve this equation for u: 2 
A 2
 2u
2
5 3

u u2
 5u   3
5 3

u u2
5 3

times u 2 on both sides  u 2  2    2 u 2
u u

2u 2  5u  3  0
 u  3 2u  1  0   u  3  0
u  3 or u 
1
2
2. Solve for x:
x 2  7 x  12
A x 2  7 x  12  0
or
 2u  1  0
```