Math 152
Chapter 1 Handout
Helene Payne
Name: Roster Notation: List elements of a set inside braces {}, separated by commas. The Set of Natural Numbers: {1, 2, 3, 4, 5, . . .} The Set of Whole Numbers: {0, 1, 2, 3, 4, 5, . . .} The Set of Integers: {. . . , −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, . . .} Set-Builder Notation: Elements of a set are not listed but described. n a o The Set of Rational Numbers: a and b are integers and b 6= 0 b ∈ - is an element of ∈ / - is not an element of 3 ∈ {1, 2, 3, 4, 5} 1 ∈ / {1, 2, 3, 4, 5} 3 50 ∈ {x|x is an integer} −5 ∈ {x|x is a rational number} 1. Use the roster method to list the following sets: (a) The set of whole numbers less than 8. { } (b) The set of integers greater than −3 and less or equal to 5. { } 2. Write the following sets using set-builder notation. (a) The set of real numbers greater than 2. { (b) The set of real numbers greater than or equal to 4 and less than 7. {
Math 152
Chapter 1 Handout
Helene Payne
Equations versus Expressions Equations have an equal sign in the middle and two sides! When we solve an equation, the equal sign stays in the middle and we work top to bottom like an elevator. Example: 4x − 17 4x − 17 + 17 4x 4x 4 x
= −5 = −5 + 17 = 12 12 = 4 = 3
Expressions do not have an equal sign! When we simplify expression, we should use an equal sign between each step. Example:
= = = =
4x − 17 + x − 5 + 2 4x + x − 17 − 5 + 2 5x − 17 − 5 + 2 5x − 22 + 2 5x − 20
3. Equation or Expression? Circle One (a) 2x − 5 + 7 − 7x2 equation/expression 5x − 3 (b) equation/expression x+2 (c) x2 + y 2 + z 2 = 5 + x + y equation/expression (d) 3x = 2 √ (e) x + 5
equation/expression equation/expression
Page 2
Math 152
Chapter 1 Handout
Helene Payne
4. Write the phrase as an algebraic expression. Let x represent the unknown number. Addition + sum plus increased by more than
Subtraction - Multiplication ×,· difference product minus times decreased by of (with fractions, percent less than twice
(a) The sum of a number and twelve.
(b) The difference of a number and ten.
(c) Twenty less than twice a number.
(d) The quotient of eight times a number and five.
Page 3
Division ÷,/ quotient divide per ratio
Math 152
Chapter 1 Handout
5. Evaluate and/or simplify the expressions below. ORDER OF OPERATIONS P = parenthesis, absolute values E = exponents, roots DM = division and multiplication from left to right SA = subtraction and addition from left to right 3 (a) − − 5
−3 4
(b) −24
(c) (−2)4
(d) −(−3)3
(e) −|22 − 7|
(f) 10 − 4(2)2 + 18 ÷ 2 · 3
7 · 32 − | − 12| ÷ 4 (g) 6(| − 7| − 2) Page 4
Helene Payne
Math 152
Chapter 1 Handout
52 + 4 ÷ 2 (h) (6 − 9)(8 − 12) ÷ 2 8 − 92 + 8
6. Simplify each expression by combining like terms. (a) 4(2y − 1) − 2(3y + 7)
(b) 18x2 + 4 − 2[5x(x2 − x) + 5]
7. Evaluate the following expressions for x = 3 and y = −1 (a) 3x − 2y − 12
(b) x2 − y 2 + 3
Page 5
Helene Payne
Math 152
Chapter 1 Handout
8. Solve each equation. (a) 2x + 7 = 11
(b)
x 3x = +5 2 4
(c)
x+1 1 2−x = + 4 6 3
(d) 0.5x − (0.2x − 0.8) = 3.5
Page 6
Helene Payne
Math 152
Chapter 1 Handout
Helene Payne
9. Solve each equation. Is it (i) conditional (one solution) (ii) an identity (an infinite number of solutions), or (iii) a contradiction (no solution)? (a) 3x − 9 = x + 3
(b) 3(x − 5) + 3 = 2x + 5 + x
(c) 4y − 5 = 2(y − 3) + 1 + 2y
Page 7
Math 152
Chapter 1 Handout
Helene Payne
10. Write the following sentence using mathematical symbols, then solve the equation using the 5 steps. The quotient of y and two is equal to eight subtracted from twice y.
11. Solve the formula for the area of a box with length, l, width w and height h: A = 2lw + 2lh + 2wh for h.
12. Find the (i) opposites and (ii)reciprocals of the numbers below: (a) 5 (i) opposite: 7 (b) − (i) opposite: 13 (c) 0 (i) opposite:
(ii) reciprocal:
(d) 1000(i) opposite:
(ii) reciprocal:
(ii) reciprocal: (ii) reciprocal:
Page 8
Math 152
Chapter 1 Handout
Helene Payne
The Laws of Exponents A. The Product Rule: am · an = am+n am B. The Quotient Rule: n = am−n , a 6= 0 a C. The Zero Exponent Rule: a0 = 1, a 6= 0 D. The Negative Exponent Rule: a−n =
1 , a 6= 0 an
E. The Power Rule: (am )n = am·n F. The Power of a Product: (ab)m = am bm G. The Power of a Quotient:
a n b
an = n , b 6= 0 b
13. Simplify the expressions below using the Laws of Exponents. Answer with positive exponents. (a) b7 · b4
(b) (4x3 )(3x4 )
y 11 (c) 5 y
Page 9
Math 152
Chapter 1 Handout
48x13 y 12 (d) 8x8 y 9
(e) 1000
(f) (5xy 21 )0
(g) 7x−7 y 5
(h) 6x6 y −2 z −1
(i)
4m2 n5 p3
2
Page 10
Helene Payne
Math 152
Chapter 1 Handout
Helene Payne
14. Write each number in decimal notation without the use of exponents. (a) 6.1 × 106
(b) 3.8 × 10−3
(c) −8.3214 × 10−4
15. Write each number in scientific notation. (a) 3500
(b) 6, 210, 000
(c) 0.0000000239
Page 11
Math 152
Chapter 1 Handout
Helene Payne
16. Perform the computations below. Answer in scientific notation! 3.6 × 108 (a) 3 × 104
(b) (5 × 105 )(1.1 × 10−1 )
(c)
0.0000072 0.0000000008
(d) (120, 000, 000)(0.0000009)
Page 12
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