Download sol review notes 2

Alg2-SOL-Equations/Inequalities
SOL A2.4 and A2.5
AII.4a: Absolute Value Equations and Inequalities
Solving Absolute Value Equations: Set up 2 cases and solve for both: Ex. 2 x  5  3
2x  5  3
2 x  5  3
2x  8
2 x  2 CHECK your solutions!
x 1
x4
Solving Absolute Value Inequalities
AND ( , < ) (graph is ‘in between’) versus OR (  , > ) (graph in opposite directions)
Set up 2 cases—Remember to flip the sign when setting up the 2nd negative case!
Ex 1: 2 x  5  3
Ex 2: 2 x  5  3
2x  5  3
2 x  5  3
2x  8
x4
2x  5  3
2x  2
and
2x  8
x 1
x4
1 x  4
A) 3 2 x  7  2  17
2 x  5  3
B)
2x  2
or
x 1
1
x  8  3  16
2
AII.4b Solving Quadratic Equations:
To solve a quadratic equation you may be asked to find the solutions, zeros, or roots. These
answers will also be found on a graph (called a parabola) as x-intercepts.
Note: A quadratic equation can have two solutions, one solution (a double root-touches the xaxis and turns around) or no real solutions (graph does not cross the x-axis).
Solving by Factoring:
1) Get the equation equal to zero. Move everything to left side.
2) Factor the left side using an appropriate technique we have learned.
3) Set each factor = 0 and solve.
EX) Solve for x: x 2  3x  10  0
C) Solve for x: 2x 2  9x  7
( x  5)( x  2)  0
x5  0 x 2  0

x5
x  2

b  b2  4ac
2a
Get the equation equal to zero. Move everything to left side. Find a, b, c and plug into
the formula: Don’t forget to simplify!
Solve for x : 3x 2  5 x  9  0
Solving using the Quadratic Formula: x 
(5)  (5) 2  4(3)(9)
EX) 
2(3)

D) 2 x 2  6 x  7  0
5  83 5  i 83

6
6
Solving by Graphing (using your calculator):
1.)
Rewrite equation in general form (set equal to 0).
2.)
Type left side of equation into the calculator for Y1
3.)
2nd TRACE; 2:ZEROS; Move the cursor to the left of one of the zeros. Press
enter. Move the cursor to the right of the zero (x-intercept). Press ENTER. Press
ENTER again. The given value for x is the solution (also known as the zero, root,
or x-intercept).
4.)
Repeat Step 3 for any other x-intercepts on the graph.
Examples: Solve by graphing (approximate zeros to the nearest hundredth if necessary).
EX) x2 + 3x – 18 = 0
E) x2 – 6x = 5
x  6,3 Factors are: ( x  6)( x  3)
II. Write the quadratic equation given the graph. Factors are (x – root)
y
EX)
F)
5
5
Roots are
-1, 3
x
Roots are
-1/2, 4
x
-5
-5
-5
5
-5
y  ( x  1)( x  3)  x 2  2 x  3
Solving Quadratics by taking the square root of both sides.
Ex)
G) x 2  72
x 2  25
x 2  25
x  5
y
5
AII.4c: Rational Equations
Step 1: Multiply each term of the equation by the LCD.
Step 2: List the values that must be excluded from the solution. These are values of the variable
that make the denominator = 0 (these values make the equation undefined).
Step 3: Solve for the variable. Check your solution in the ORIGINAL equation!
Ex.
5
3

0
p 1 p  2
LCD is (p - 1)(p + 2)
5
3

( p  1)( p
p 1 p  2
5( p  2)  3( p  1)  0
( p  1)( p  2)
 2)
H)
4 x 12 5 x 7
 

3
x 4 12
 0( p  1)( p  2)
5 p  10  3 p  3  0
2 p  13  0
2 p  13
p  13 / 2
AII.4d: Radical Equations
Steps to solving radical equations:
(a) get the radical on the side by itself (“isolate the radical”)
(b) square (or cube, etc) both sides of the equation
(c) solve for the variable
(d) check for extraneous solutions
1
5x 1  1
Ex. 2 3x  1  10
I) 2 
3
3x  1  5

3x  1

2
3x  1  25
x 8
  5
2
AII.5: Non-linear systems
2.
Solve linear quadratic system: y = x2 - 4x - 2 and y = x – 2
1. Enter the first equation into Y1.
2. Enter the second equation into Y2.
3. Hit GRAPH.
4. Use the INTERSECT option twice to find the two
locations where the graphs intersect (the answers).
2nd TRACE (CALC) #5 intersect
Move spider close to the intersection.
Hit ENTER 3 times.
5. Answer: (5,3) and (0,-2)
and
J) Use your calculator to approximate the solutions:
y  x2  2x  6
(Don’t forget to solve for y first!)
3x  y  12
K) How many solutions would each of the following systems have?
Give approximate solutions.
y
7
6
5
4
3
2
1
–6 –5 –4 –3 –2 –1
–1
1
2
3
4
5
6
x
–2
–3
–4
–5
–6
–7
# of solutions: _____
# of solutions: ____
Approx. solutions: ____________________
Approx. solutions: ____________________