exam 1 Flashcards
sections 8-9 review
[8.1] Solve for:
3 + | x | = -1
No solution/contradiction. An absolute value can not be negative.
[8.1] What is the first step to solving this equation?
(3/5x - 1/2) = (2/3x + 3/4)
Finding the least common denominator.
[8.1] Explain how you would solve this.
|x + 5| = |3x - 7|
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)
[8.2] Explain the difference between a “trap” and an “or.”
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
[8.2] Solve for:
|x + 3| > - 5
All real numbers. An absolute value, which will always be positive, is always greater than a negative.
[8.2] Solve.
|x| = 3
x = - 3 OR x=3
[8.2] Solve.
|x| < 3
Trap equation.
-3 < x < 3
[8.2] Solve.
|x| > 3
“Or” equation.
x < -3 OR x > 3
[8.2] Solve.
|x| < - 3
No solution.
[8.2] Solve.
|x| > - 3
All real numbers.
[8.3] How do you find the x-intercept and the y-intercept of this equation?
x + 3y = 6
For the x-intercept, replace y with 0.
For the y-intercept, replace x with 0.
[8.3] What is the slope-intercept form equation for graphing?
y = mx + b
[8.3] What do the “m” and “b” represent in y = mx + b?
The “m” stands for the slope and “b” stands for the y-intercept.
[8.3] When given this slope formula “y = - 5/2x + 5” how does the negative affect the slope?
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.
[8.3] What is the “run” in this slope?
f(x) = - 5x + 8
The run is 1, since the rise is a whole number, the denominator is 1.
[8.3] What is special about this slope formula?
f(x) = |x - 5|
It has a “v” shape.
[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?
Vertex.
[8.3] What is the formula for a “V” graph? (absolute value)
f(x) = |x - h| + k
[8.3] In this formula, “|x - h| + k” what do the h and k stand for?
The vertex. (h, k)
EX: f(x) = |x - 2| - 4
vertex = ( 2, -4 )
[8.4] When factoring, if possible, the first step is…
to factor out the GCF. (Greatest Common Factor)
[8.4] After determining the number of terms in the polynomial, if there are 4 terms…
Factor by grouping.
[8.4] After determining the number of terms in the polynomial, if there are 3 terms…
Diamond or AC method.
Diamond: x^2 + bx + c
AC: ax^2 + bx + c
[8.4] After determining the number of terms in the polynomial, if there are 2 terms…
(HINT: 4 formulas)
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)
[8.5] What is the best method for the following, elimination or substitution?
2x + y = 7
3x + 4y = 8
Elimination.
[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
Elimination.
[8.5] What is the best method for the following, elimination or substitution?
x = 7 - 2y
5x - 2y = 20
Substitution.
[9.1] What are the solutions for x?
x^2 = 49
7 and -7.
[9.1] What is the square root rule of fractions?
The square root of a divided by b, is equal to the square root of a divided by the square root of b.
[9.1] Solve.
The fifth root of -32.
Since the “nth” root is odd, in this case 5, it can produce a real solution.
[9.1] What is the product rule for radicals?
Nth root of a multiplied by nth root of b is equal to the nth root of ab.
[9.1] What is the product rule for same radicals?
Example, square root of 2 multiplied by square root of 2.
Radicals with the same radican (number inside) are equal to the number inside the radical.
[9.1] What is the quotient rule for dividing radicals?
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)
[9.2] What is the rational exponent rule?
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.
[9.2] x^m times x^n =
x ^ m + n
[9.2] ( x ^ m ) ^ n =
x ^ m times n
[9.2] x ^ m / x ^ n =
x ^ m - n
[9.2] ( xy ) ^ n =
x ^ n times y ^ n
[9.2] ( x / y ) ^ n =
x ^ n / y ^ n
[9.2] x ^ 0 =
1
[9.2] x ^ -n
1 / x ^ n
[9.3] Simplify this expression.
Cube root of x to the 5th power.
x times cube root of x to the 2nd power
[9.4] When rationalizing the denominator, what are the two rules when dealing with radicals?
- There can be no fractions inside radicals.
- There can be no radicals in the denominator.
[9.5] We must check out solutions if we _______ both sides of an equation to an ________ power or exponent. This can introduce extraneous solutions.
raise, even
[9.6] What are the 2 rules to how we define “i”.
The square root of -1 = i.
i squared = -1.
[9.6] What is the complex numbers expression? (The correct format in writing “i” equations simplified)
a + bi.
