Homework LogThurs & Fri
10/22
Lesson Rev
Learning Objective: To remember everything in Ch 2
Hw: #216 Pg. 155 #1 – 85 odd
Homework LogMon
10/24
Lesson Rev
Learning Objective: To remember everything in Ch 2
Hw: Extra Credit Test Review
10/22/15 Chapter 2 Review
Advanced Math/Trig
Learning Objective
To remember everything in Chapter 2!
Solve a Rational Equation
1.
x + 3(1 – x) = –1(1 – x)
x + 3 – 3x = –1 + x
3 – 2x = –1 + x
–3x = –4
x
LCD:x(1 – x)
2. – 7 – 7
– 4 – 4
2x – 3 = 4 or 2x – 3 = – 4
2x = 7 2x = –1x = or x =
Absolute Value Equations
Warm–up #2 Solutions3. Solve x + 5 = 3x – 2 or x + 5 = –(3x – 2)–2x = –7 or x + 5 = –3x + 2 4x = –3x = or x = Nope!Check for extraneous solutions!
Solving for a Variable4. Solve for S:
PS + PF = S – PS – PS
PF = S – PS PF = S(1 – P)
(S + F)
(S + F)
Simple Interest5. Part of $20,000 is to be invested at 15% and the remainder at 9%. How much should be invested at each rate to yield an annual interest income of $2520.
Principal rate time Interest
Inv 1
Inv 2
Total 20000
x20000 – x
.15.09
11
• • =.15x
.09(20000 – x)2520
equation!.15x + .09(20000 – x) = 2520
.15x + .09(20000 – x) = 2520 .15x + 1800 – .09x = 2520 .06x = 720 x = 12000
20000 – 12000 = 8000
$12,000 at 15%$8,000 at 9%
Simple Interest #5 cont’d
Investment6. If $9000 is invested at 7% per year, how much additional money needs to be invested at 14% per year so that the total annual interest income from the investments is $1330?Principal rate time Interest
Inv 1
Inv 2
Total 9000 + x
9000x
.07.14
11
• • =9000(.07) = 630
.14x1330
equation!630 + .14x = 1330
630 + .14x = 1330.14x = 700
x = 5000
$5,000 at 14%
Investment Cont’d
Mixture7. I want to dilute 40 L of a solution that is 80% acid to one that is 50% acid. How much water should be added to the acid solution?
Amount % Total
Solution 1
Solution 2
Mix 50(40 + x)
80
40 + x
x
40
0
50
40(80) = 32000
3200 = 50(40 + x)
• =24 L
Distance Problem8. Laura & Luke left school at the same time and went in opposite directions. Laura was driving 40 mph faster than Luke. After 3 hours, they were 330 miles apart. How fast was Laura driving?Rate Time Distance
Laura
Luke
Total
3
x
x + 40
3
3(x + 40)
3x
3(x + 40) + 3x = 3303x + 120 + 3x = 330x = 35
• =330
75 mph
Drain/Work Problem9. An Olympic sized pool can be filled by pipe A in 12 hours and by pipe B in 10 hours. There is also a drain pipe that drains the entire pool in 6 hours. If the valves of pipe A, pipe B and the drain pipe are open, how long will it take to fill the pool?
AloneRate
Time Together
Part of Job
Completed
Pipe A
Pipe B
Drain Pipe
x
x
• =x
𝟏𝟏𝟐𝟏𝟏𝟎
−𝟏𝟔
𝒙𝟏𝟐𝒙𝟏𝟎
−𝒙𝟔
Pipe A’s part + Pipe B’s part + Drain’s part= 1 Whole Job Completed(60) (60) (60)(60)
x = 60 hours
Drain/Work Problem #9 Cont’d
AloneRate
Time Together
Part of Job
Completed
Pipe A
Pipe B
Drain Pipe
x
x
• =x
𝟏𝟏𝟐𝟏𝟏𝟎
−𝟏𝟔
𝒙𝟏𝟐𝒙𝟏𝟎
−𝒙𝟔
𝑥12
+𝑥
10−𝑥6
=1
5x + 6x – 10x = 60
Work Problem10. Working together, Scott and Jenna can sweep a porch in 10 minutes. If Jenna worked alone, it would have taken her 15 minutes. How long does it take Scott to sweep the porch alone?Alone
RateTime
TogetherPart of
JobComplete
d
Scott
Jenna
10
10
Scott’s part + Jenna’s part = 1 Whole Job Completed
• =𝟏𝟏𝟓
𝟏𝒙
𝟏𝟎𝒙𝟏𝟎𝟏𝟓
Work Problem #10 Cont’d
10𝑥
+1015
=1(1 5𝑥 ) (1 5𝑥 )
(1 5𝑥 )
150 + 10x = 15x150 = 5x
30 minutes
AloneRate
Time Together
Part of Job
Completed
Scott
Jenna
10
10
Scott’s part + Jenna’s part = 1 Whole Job Completed
• =𝟏𝟏𝟓
𝟏𝒙
𝟏𝟎𝒙𝟏𝟎𝟏𝟓
11. Less ThAND
2x – 5 < 3 and 2x – 5 > – 3 + 5 + 5 +5 +5 2x < 8 2x > 2 2 2 2 2x < 4 and x > 1 1 < x < 4
Solve Absolute Value Inequalities
(1, 4)
12. GreatOR
5x + 3 > 7 or 5x + 3 < – 7 – 3 – 3 – 3 – 35x > 4 5x < – 10 5 5 5 5x > 4/5 or x < – 2
Solve Absolute Value Inequalities
(–, –2) (, )
13. – 5
All Real Numbers
Absolute Value is always positive & will ALWAYS be greater than a negative number!!
Solve Absolute Value Inequalities
(–, )
14. 20
No Solution
Absolute Value is always positive & will NEVER be less than a negative number!!
Solve Absolute Value Inequalities
∅
Solve by Factoring15.
x(2x – 1)(x + 3) = 0
x = 0 x + 3 = 0 2x – 1 = 0
{–3, 0, }
2 #s that mult to
–6
5& add to
6 –1
Solve by Completing the Square
16. = 0
– 20 – 20
= – 20 +
x – 4 =
√❑√❑
16 16
=
Or
Solve by Completing the Square
17.
4 4 4
+
√¿¿
❑¿
√¿¿
❑¿
Solve by Factoring18.
u(u – 7) + 2(u – 7) = 0
(u – 7)(u + 2) = 0
Let
2 #s that mult to–14
–5& add to
–7 2
Solve by Factoring18. (u – 7)(u + 2) = 0
((x +2) – 7)((x +2) + 2) = 0
(x – 5)(x + 4) = 0
x – 5 = 0 x + 4 = 0
{– 4, 5}
Now replace u with
2 Answers!!
Highest Power is 2!!
Sum – Product Rule
x2 – (sum)x + product = 0
sum: 5 + –3 = 2
product: (5)(–3) = –
15x2 – (2)(x) + (–15) = 0
x2 – 2x – 15 = 0
19. Find a monic quadratic eq’n whose roots are 5 & –3
2 4
2
b b acx
a
Quadratic Formula
Xavier is a negative boy who couldn’t decide (yes or no) whether to go to a radical party.
It turns out that this boy is a total square because he missed out on 4 awesome chicks.
And the party was all over at 2 AM.
2Given: 0ax bx c
DiscriminantDiscriminant – tells the nature of the roots
(Part under the radical)
Discriminant Roots
Zero 1 real double root
Positive 2 real roots
Negative 2 imaginary roots