exam 1

Question 1

[8.1] Solve for:
3 + | x | = -1

Answer 1

No solution/contradiction. An absolute value can not be negative.

Question 2

[8.1] What is the first step to solving this equation?
(3/5x - 1/2) = (2/3x + 3/4)

Answer 2

Finding the least common denominator.

Question 3

[8.1] Explain how you would solve this.
|x + 5| = |3x - 7|

Answer 3

You would first set both equations equal to each other without any changes, both sides stay the same.

x + 5 = 3x - 7

Then, you would set the second equation with both sides equal to each other but one side is set as negative and the other positive.

x + 5 = - (3x - 7)

Question 4

[8.2] Explain the difference between a “trap” and an “or.”

Answer 4

A “trap” equation points towards the left side and is called a trap because you trap your x-value by adding another value to the left side.

EX: |x + 2| < 3
- 3 < x + 2 < 3

An “or” equation points towards the right side and your x-value must equal the opposite of the original equation OR the original equation as is.

EX: |x + 6| > 8

x + 6 < -8 OR
x + 6 > 8

Question 5

[8.2] Solve for:
|x + 3| > - 5

Answer 5

All real numbers. An absolute value, which will always be positive, is always greater than a negative.

Question 6

[8.2] Solve.
|x| = 3

Answer 6

x = - 3 OR x=3

Question 7

[8.2] Solve.
|x| < 3

Answer 7

Trap equation.
-3 < x < 3

Question 8

[8.2] Solve.
|x| > 3

Answer 8

“Or” equation.
x < -3 OR x > 3

Question 9

[8.2] Solve.
|x| < - 3

Answer 9

No solution.

Question 10

[8.2] Solve.
|x| > - 3

Answer 10

All real numbers.

Question 11

[8.3] How do you find the x-intercept and the y-intercept of this equation?

x + 3y = 6

Answer 11

For the x-intercept, replace y with 0.
For the y-intercept, replace x with 0.

Question 12

[8.3] What is the slope-intercept form equation for graphing?

Answer 12

y = mx + b

Question 13

[8.3] What do the “m” and “b” represent in y = mx + b?

Answer 13

The “m” stands for the slope and “b” stands for the y-intercept.

Question 14

[8.3] When given this slope formula “y = - 5/2x + 5” how does the negative affect the slope?

Answer 14

Since the slope is -5/2, the rise being -5 and the run being 2, the negative applies to the rise making it slope downward instead of up.

Question 15

[8.3] What is the “run” in this slope?
f(x) = - 5x + 8

Answer 15

The run is 1, since the rise is a whole number, the denominator is 1.

Question 16

[8.3] What is special about this slope formula?
f(x) = |x - 5|

Answer 16

It has a “v” shape.

Question 17

[8.3] When a slope formula contains an absolute value, it has a special “v” shape when graphed, what is the bottom of the “v” called?

Answer 17

Vertex.

Question 18

[8.3] What is the formula for a “V” graph? (absolute value)

Answer 18

f(x) = |x - h| + k

Question 19

[8.3] In this formula, “|x - h| + k” what do the h and k stand for?

Answer 19

The vertex. (h, k)

EX: f(x) = |x - 2| - 4
vertex = ( 2, -4 )

Question 20

[8.4] When factoring, if possible, the first step is…

Answer 20

to factor out the GCF. (Greatest Common Factor)

Question 21

[8.4] After determining the number of terms in the polynomial, if there are 4 terms…

Answer 21

Factor by grouping.

Question 22

[8.4] After determining the number of terms in the polynomial, if there are 3 terms…

Answer 22

Diamond or AC method.
Diamond: x^2 + bx + c
AC: ax^2 + bx + c

Question 23

[8.4] After determining the number of terms in the polynomial, if there are 2 terms…
(HINT: 4 formulas)

Answer 23

Difference of squares: a^2 - b^2 = (a - b)(a + b)

Sum of squares: a^2 + b^2 = PRIME

Difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Question 24

[8.5] What is the best method for the following, elimination or substitution?

2x + y = 7
3x + 4y = 8

Answer 24

Elimination.

Question 25

[8.5] What is the best method for the following, elimination or substitution?

x + 3y - 4z = -12
-2x + 9y + 3z = -5
5x - 7y + z = -20

Answer 25

Elimination.

Question 26

[8.5] What is the best method for the following, elimination or substitution?

x = 7 - 2y
5x - 2y = 20

Answer 26

Substitution.

Question 27

[9.1] What are the solutions for x?

x^2 = 49

Answer 27

7 and -7.

Question 28

[9.1] What is the square root rule of fractions?

Answer 28

The square root of a divided by b, is equal to the square root of a divided by the square root of b.

Question 29

[9.1] Solve.

The fifth root of -32.

Answer 29

Since the “nth” root is odd, in this case 5, it can produce a real solution.

Question 30

[9.1] What is the product rule for radicals?

Answer 30

Nth root of a multiplied by nth root of b is equal to the nth root of ab.

Question 31

[9.1] What is the product rule for same radicals?

Example, square root of 2 multiplied by square root of 2.

Answer 31

Radicals with the same radican (number inside) are equal to the number inside the radical.

Question 32

[9.1] What is the quotient rule for dividing radicals?

Answer 32

Nth root of a divided by the nth root of b is equal to the nth root of a divided by b. This only applies if the index is the same. (AKA the root outside of the box)

Question 33

[9.2] What is the rational exponent rule?

Answer 33

x to the m/nth power (exponent) is equal to the nth root of x to the m power.
the denominator of the exponent is always the index of the radical.

Question 34

[9.2] x^m times x^n =

Answer 34

x ^ m + n

Question 35

[9.2] ( x ^ m ) ^ n =

Answer 35

x ^ m times n

Question 36

[9.2] x ^ m / x ^ n =

Answer 36

x ^ m - n

Question 37

[9.2] ( xy ) ^ n =

Answer 37

x ^ n times y ^ n

Question 38

[9.2] ( x / y ) ^ n =

Answer 38

x ^ n / y ^ n

Question 39

[9.2] x ^ 0 =

Answer 39

1

Question 40

[9.2] x ^ -n

Answer 40

1 / x ^ n

Question 41

[9.3] Simplify this expression.

Cube root of x to the 5th power.

Answer 41

x times cube root of x to the 2nd power

Question 42

[9.4] When rationalizing the denominator, what are the two rules when dealing with radicals?

Answer 42

1. There can be no fractions inside radicals.
2. There can be no radicals in the denominator.

Question 43

[9.5] We must check out solutions if we _______ both sides of an equation to an ________ power or exponent. This can introduce extraneous solutions.

Answer 43

raise, even

Question 44

[9.6] What are the 2 rules to how we define “i”.

Answer 44

The square root of -1 = i.
i squared = -1.

Question 45

[9.6] What is the complex numbers expression? (The correct format in writing “i” equations simplified)

Answer 45

a + bi.