Solving Linear Equations and Inequalities
1. An ________ is a mathematical sentence with an equal sign. There are expressions on the left and right signs of the equal sign.
2. To solve an equation, we find the value of the variable that makes the equation true. This value is called the __________.
3. An ___________ is a mathematical sentence that contains a sign that indicates that the values on each side make a non equal comparison.
1. An EQUATION is a mathematical sentence with an equal sign. There are expressions on the left and right signs of the equal sign.
2. To solve an equation, we find the value of the variable that makes the equation true. This value is called the SOLUTION.
3. An INEQUALITY is a mathematical sentence that contains a sign that indicates that the values on each side make a non equal comparison.
1. A _________ is an expression that has 1 term. For example: 38m
2. A _________ is an expression that has 2 terms. For example: -7y + 1/2
3. A _________ is an expression that has 3 terms. For example: 8a^2 - 3/5ab +6b^2
4. A _____________ is an expression of more than two algebraic terms that is the sum (or difference) of several terms that contain different powers of the same variable(s).
1. A MONOMIAL is an expression that has 1 term. For example: 38m
2. A BINOMIAL is an expression that has 2 terms. For example: -7y + 1/2
3. A TRINOMIAL is an expression that has 3 terms. For example: 8a^2 - 3/5ab +6b^2
4. A POLYNOMIAL is an expression of more than two algebraic terms that is the sum (or difference) of several terms that contain different powers of the same variable(s).
Simplify the following radicals:
1. √64
2. √169
3. √121/9
1. 8
2. 13
3. 11/3
A ___________ is an equation that has a variable to the second power but no variable higher than the second power. A __________ always has this form: y = ax^2 +bx +c.
A QUADRATIC EQUATION is an equation that has a variable to the second power but no variable higher than the second power. A QUADRATIC EQUATION always has this form: y = ax^2 +bx +c.
1. ____________ or Counting Numbers: The set of all positive numbers starting at 1 that have no fractional or decimal part; also called whole numbers. Examples: 1, 2, 3, 4, 5, .....
2. ____________: The set of all natural numbers and 0. Examples: 0, 1, 2, 3, 4, 5, .....
3. ____________: The set of all whole numbers, including negative natural numbers. Examples: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, .....
4. ____________: The set of all numbers that can be written by dividing one integer by another. These include any number that can be written as a fraction or ratio. Examples: 3.27, 1/2, -8, 2, .....
5. ____________: The set of all numbers that are not rational numbers. These are numbers that cannot be written by dividing one integer by another. When we write (a)n ____________ as a decimal, it goes on forever without repeating itself. Examples: 0.3471578910..., √5, π, ....
6. ____________: The set of all numbers on a number line. Real numbers include all rational and irrational numbers. This can be zero, positive or negative integers, decimals, fractions, etc. Examples: 8, -19, 0, 3/2, √47, √25, π, ....
1. NATURAL NUMBERS or Counting Numbers: The set of all positive numbers starting at 1 that have no fractional or decimal part; also called whole numbers. Examples: 1, 2, 3, 4, 5, .....
2. WHOLE NUMBERS: The set of all natural numbers and 0. Examples: 0, 1, 2, 3, 4, 5, .....
3. INTEGERS: The set of all whole numbers, including negative natural numbers. Examples: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, .....
4. RATIONAL NUMBERS: The set of all numbers that can be written by dividing one integer by another. These include any number that can be written as a fraction or ratio. Examples: 3.27, 1/2, -8, 2, .....
5. IRRATIONAL NUMBERS: The set of all numbers that are not rational numbers. These are numbers that cannot be written by dividing one integer by another. When we write (a)n IRRATIONAL NUMBER as a decimal, it goes on forever without repeating itself. Examples: 0.3471578910..., √5, π, ....
6. REAL NUMBERS: The set of all numbers on a number line. Real numbers include all rational and irrational numbers. This can be zero, positive or negative integers, decimals, fractions, etc. Examples: 8, -19, 0, 3/2, √47, √25, π, ....
Solve the following linear equation for x.
3(x - 8) + 4x = 8x + 4
x = -28
Factor:
1. 3x^2 + 15x - 42
2. 5x^3 - 20x^2 - 60x
1. 3(x + 7)(x - 2)
2. 5x(x + 2)(x - 6)
Simplify:
1. √45
2. √300
3. √72
1. 3√5
2. 10√3
3. 6√2
For the following problems, state the values of a, b, and c, and if it is a quadratic equation or not. (If it is not a quadratic equation there is no a, b, or c.)
1. y = x^2 + 3x - 15
2. y = 6x^5 - 0.7x + π
3. y = -1/3x^2 + 4
4. y = 9x^2
5. y = 8x^2 - 2x + 9
1. Yes, it is a quadratic equation. a = 1 b = 3 c = -15
2. No, it is not a quadratic equation.
3. Yes, it is a quadratic equation. a = -1/3 b = 0 c = 4
4. No, it is not a quadratic equation.
5. Yes, it is a quadratic equation. a = 8 b = -2 c = 9
For which value of P and W is P + W a rational number?
1) P = 1/√3 and W= 1/√6
2) P = 1/√4 and W = 1/√9
3) P = 1/√6 and W = 1/√10
4) P = 1/√25 and W = 1/√2
Answer: 2)
The value of x which makes 2/3 (1/4x - 2) = 1/5 (4/3x - 1) true is _________.
x = -11.3 (repeating)
Factor:
1. 121 - x^2
2. x^2 - 16
1. (11 - x)(11 + x)
2. (x + 4)(x - 4)
Is the product of √1024 and -3.4 rational or irrational? Explain your reasoning.
Rational, as √1024 x - 3.4 = 32 x - 3.4 = - 108.8, which is the ratio of two integers, - 1088/10.
Solve the equation: 2x^2 - 11x = -12
Answer: x = 3/2 x = 4
Given the following expressions:
I. - 5/8 + 3/5
II. 1/2 + √2
III. (√5) x (√5)
IV. 3 x (√49)
Which expression(s) result in an irrational number?
Answer: II.
Joy wants to buy strawberries and raspberries to bring to a party. Strawberries cost $1.60 per pound and raspberries cost $1.75 per pound. If she only has $10 to spend on berries, which inequality represents the situation where she buys x pounds of strawberries and y pounds of raspberries?
Answer: 1.60x + 1.75y ≤ 10
The expression (-2a^2 b^3 )(4ab^5 )(6a^3 b^2 ) is equivalent to ___________.
Answer: -48a^6 b^10
The sum of √75 and √3 is _________. (Round to the nearest hundredth if needed.)
The sum of √75 and √3 is
6√3 or 10.39
Solve for x using the quadratic formula: 4x^2 - x - 6 = 0
x = 1 ∓ √97 / 8
Which statement is not always true?
1) The product of two irrational numbers is irrational.
2) The product of two rational numbers is rational.
3) The sum of two rational numbers is rational.
4) The sum of a rational number and an irrational number is irrational.
Answer: 1)
An animal shelter spends $2.35 per day to care for each cat and $5.50 per day to care for each dog. Pat noticed that the shelter spent $89.50 caring for cats and dogs on Wednesday. Write an equation to represent the possible numbers of cats and dogs that could have been at the shelter on Wednesday. Pat said that there might have been 8 cats and 14 dogs at the shelter on Wednesday. Are Pat’s numbers possible? Use your equation to justify your answer. Later, Pat found a record showing that there were a total of 22 cats and dogs at the shelter on Wednesday. How many cats were at the shelter on Wednesday?
2.35c + 5.50d = 89.50
Pat’s numbers are not possible:
2.35(8) + 5.50(14) ≠ 89.50
18.80 + 77.00 ≠ 89.50
95.80 ≠ 89.50
c + d = 22 2.35c + 5.50(22 − c) = 89.50
2.35c + 121 − 5.50c = 89.50
−3.15c = −31.50
c = 10
Simplify: 27k^5 m^8 / (4k^3 )(9m^2 )
Answer: 3k^2 m^6 / 4
Darin simplified 5√5 + 2√5 and got 15.7. Martha simplified the same expression and got 50. Who is correct?
Answer:
Darin is correct.
What is the quadratic formula?
________________________
x = -b ∓ √b^2 - 4ac / 2a
Ms. Fox asked her class "Is the sum of 4.2 and √2 rational or irrational?" Patrick answered that the sum would be irrational. State whether Patrick is correct or incorrect. Justify your reasoning.
Answer: The sum of an irrational and a rational number is irrational. Therefore, Patrick is correct.