College Algebra Foundations: Operations, Properties, and Problem Solving

Review of Algebra

Translating Words to Algebraic Expressions

Understanding how to translate verbal statements into algebraic expressions is a foundational skill in algebra. Key words and phrases indicate specific operations.

• Addition: "added to," "sum of," "plus," "more than," "increased by"

• Subtraction: "subtracted," "minus," "difference," "less than," "decreased by"

• Multiplication: "multiplied by," "product of," "times," "twice," "of"

• Division: "divided by," "quotient of," "ratio of," "per"

Examples:

• "A number increased by 6" → $x + 6$

• "7 more than a number" → $x + 7$

• "The sum of two variables" → $x + y$

• "A number less than 3" → $3 - x$

• "9 decreased by a number" → $9 - x$

• "The product of 8 and a number" → $8x$

• "3 times the cost of an item" → $3c$

• "The quotient of x and 4" → $\frac{x}{4}$

• "The ratio of 3 and 5" → $\frac{3}{5}$

Example: Translate

• "Twice a number" → $2x$

• "Thirteen less than one quarter of some number" → $\frac{1}{4}x - 13$

Evaluating Expressions and Order of Operations

To evaluate algebraic expressions, substitute the given values and follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

• Example 1: $3(x - 7) + 2$, for $x = 10$ Substitute: $3(10 - 7) + 2 = 3(3) + 2 = 9 + 2 = 11$

• Example 2: $12 + 3(n + 2)^2$, for $n = 1$ Substitute: $12 + 3(1 + 2)^2 = 12 + 3(3)^2 = 12 + 3(9) = 12 + 27 = 39$

• Example 3: $[10 - (a - b)]^2$, for $a = 7$, $b = 2$ Substitute: $[10 - (7 - 2)]^2 = [10 - 5]^2 = 5^2 = 25$

• Example 4: $[5(r + s)]^2$, for $r = 1$, $s = 2$ Substitute: $[5(1 + 2)]^2 = [5(3)]^2 = 15^2 = 225$

Absolute Value

The absolute value of a number $a$, written $|a|$, is its distance from 0 on the number line.

• $|{-5}| = 5$

• $|5| = 5$

• $-|{-15}| = -15$

• $|0| = 0$

Operations with Real Numbers

Rules for addition and subtraction depend on the signs of the numbers involved.

• Adding positive numbers: Add as usual; result is positive.

• Adding negative numbers: Add absolute values; result is negative.

• Adding numbers with different signs: Subtract the smaller absolute value from the larger; use the sign of the number with the larger absolute value.

Examples:

• $3 + 5 = 8$

• $-3 + (-5) = -8$

• $3 + (-5) = -2$

• $-3 + 5 = 2$

Removing Parentheses with Double Signs

When simplifying expressions, use these rules:

• $+(+) = +$

• $+(-) = -$

• $-(-) = +$

• $-(+) = -$

Examples:

• $6 + (4) = 10$

• $6 - (-4) = 10$

• $6 + (-4) = 2$

• $6 - (4) = 2$

Multiplication and Division of Real Numbers

To multiply or divide two nonzero real numbers:

• If the signs are the same, the result is positive.

• If the signs are different, the result is negative.

Examples:

• $2 \times 5 = 10$

• $-2 \times -5 = 10$

• $2 \times -5 = -10$

• $-2 \times 5 = -10$

Division with Fractions: To divide by a fraction, multiply by its reciprocal.

• $\frac{2}{3} \div \frac{7}{11} = \frac{2}{3} \times \frac{11}{7} = \frac{22}{21}$

Order of Operations

Follow the order: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

• Example: $3[2 + 5(4 - 2)]$ First, $4 - 2 = 2$; then $5 \times 2 = 10$; then $2 + 10 = 12$; finally $3 \times 12 = 36$.

Distributive Law

The distributive law states $a(b + c) = ab + ac$.

• Example 1: $7(x + 1) = 7x + 7$

• Example 2: $5(m - n) = 5m - 5n$

• Example 3: $-5(2a + 3b) = -10a - 15b$

Equations & Inequalities

Solving Linear Equations

To solve linear equations, follow these steps:

1. Remove fractions or decimals.

2. Remove parentheses using the distributive law.

3. Combine like terms on each side.

4. Move all variable terms to one side, constants to the other.

5. Isolate the variable using division.

6. Check your answer by substitution.

Examples:

• $x - 2.5 = 13.4$ $x = 13.4 + 2.5 = 15.9$

• $8t = 72$ $t = \frac{72}{8} = 9$

• $\frac{2}{3}x = 30$ $x = 30 \times \frac{3}{2} = 45$

• $4a + 25 = 9$ $4a = 9 - 25 = -16$ $a = -4$

• $2y - 8 = 9$ $2y = 9 + 8 = 17$ $y = 8.5$

Combining Like Terms

Combine terms with the same variable and exponent.

• $18p - 12 + 3p + 8 = (18p + 3p) + (-12 + 8) = 21p - 4$

• $-7t^2 + 3t + 5t^3 - t^3 + 2t^2 - t = (5t^3 - t^3) + (-7t^2 + 2t^2) + (3t - t) = 4t^3 - 5t^2 + 2t$

• $2x + 3(5x - 7) = 2x + 15x - 21 = 17x - 21$

• $2(x - 3) + 4(7 - x) = 2x - 6 + 28 - 4x = (2x - 4x) + (28 - 6) = -2x + 22$

Solving Multi-Step Equations

• $4x + 5x = 63$ $9x = 63$ $x = 7$

• $4(t - 3) - t = 6$ $4t - 12 - t = 6$ $3t = 18$ $t = 6$

• $5y - (2y - 10) = 25$ $5y - 2y + 10 = 25$ $3y = 15$ $y = 5$

• $7r - 2 + 5r = 6r + 6 - 4r$ $(7r + 5r) - 2 = (6r - 4r) + 6$ $12r - 2 = 2r + 6$ $10r = 8$ $r = 0.8$

Problem Solving

The Five Steps of Problem Solving

1. Familiarize yourself with the problem.

2. Translate to mathematical language (write an equation).

3. Carry out mathematical manipulation (solve the equation).

4. Check your answer in the original problem.

5. State the answer clearly in a complete sentence.

Examples (Translation):

• "The sum of 2 numbers is 91. One of the numbers is 9 more than the other." Let $x$ = first number, $y$ = second number. $x + y = 91$, $x = y + 9$

• "The degree measures of the angles in a triangle are three consecutive integers." Let $x$, $x+1$, $x+2$ be the angles. $x + (x+1) + (x+2) = 180$

• "Find three consecutive odd integers such that the sum of the first, twice the second, and three times the third is 70." Let $x$, $x+2$, $x+4$ be the integers. $x + 2(x+2) + 3(x+4) = 70$

Examples (Solving):

• "Tess paid $124 for her calculator, which is $13 less than Tony paid. How much did Tony pay?" Let$x$= Tony's price.$x - 13 = 124x = 137$

• "The length of a rectangular mirror is three times its width, and its perimeter is 120 cm. Find the length and width." Let $w$ = width, $l = 3w$ $2l + 2w = 120$ $2(3w) + 2w = 120$ $6w + 2w = 120$ $8w = 120$ $w = 15$, $l = 45$

• "The width of a rectangular garden is one-third its length, and its perimeter is 32 m. Find the dimensions." Let $l$ = length, $w = \frac{1}{3}l$ $2l + 2w = 32$ $2l + 2(\frac{1}{3}l) = 32$ $2l + \frac{2}{3}l = 32$ $\frac{8}{3}l = 32$ $l = 12$, $w = 4$

Formulas, Models, and Geometry

Solving Formulas for a Variable

Isolate the desired variable using algebraic manipulation.

• $E = wA$, solve for $A$: $A = \frac{E}{w}$

• $V = lwh$, solve for $h$: $h = \frac{V}{lw}$

• $G = w + 150n$, solve for $n$: $n = \frac{G - w}{150}$

• $2x + 3y = 4$, solve for $y$: $3y = 4 - 2x$ $y = \frac{4 - 2x}{3}$

• $3x - 7y = 2$, solve for $y$: $-7y = 2 - 3x$ $y = \frac{3x - 2}{7}$

• $V = \frac{4}{3}\pi r^3$, solve for $r^3$: $r^3 = \frac{3V}{4\pi}$

Properties of Exponents

Exponent Rules

Rule	Formula	Example
Zero Exponent	$a^0 = 1$	$15^0 = 1$
Identity Exponent	$a^1 = a$	$7^1 = 7$
Product Rule	$a^m \cdot a^n = a^{m+n}$	$2^2 \cdot 2^3 = 2^{2+3} = 2^5$
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$	$\frac{3^5}{3^4} = 3^{5-4} = 3^1$
Power Rule	$(a^m)^n = a^{mn}$	$(x^4)^6 = x^{24}$
Power of a Product	$(ab)^n = a^n b^n$	$(a^7b^2)^3 = a^{21}b^6$
Power of a Quotient	$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$	$\left(\frac{x^2}{y^5}\right)^4 = \frac{x^8}{y^{20}}$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-3} = \frac{1}{x^3}$

Examples with Exponents

• Multiply and simplify: $5x^4 \cdot 4x^3 = 20x^{4+3} = 20x^7$

• $(m^5 n^2)(m^3 n p^0) = m^{5+3} n^{2+1} p^0 = m^8 n^3$ (since $p^0 = 1$)

• Divide and simplify: $\frac{m^7 n^5}{m^2 n^4} = m^{7-2} n^{5-4} = m^5 n^1$

• Write without negative exponents: $t^{-9} = \frac{1}{t^9}$

• $3a^8 b^{-6} = 3a^8 \frac{1}{b^6} = \frac{3a^8}{b^6}$

• $\frac{1}{x^7}$ (already no negative exponents)

• Simplify: $6^{-3} \cdot 6^{-5} = 6^{-8} = \frac{1}{6^8}$

• Simplify: $(t^{-8})^{-5} = t^{40}$

• Simplify: $\left(\frac{m^2 n^{-1}}{4}\right)^3 = \frac{m^6 n^{-3}}{64} = \frac{m^6}{64 n^3}$

Scientific Notation

Definition and Conversion

Scientific notation expresses numbers as $N \times 10^m$, where $1 \leq N < 10$ and $m$ is an integer.

• Convert $4.92 \times 10^3$ to decimal: $4,920$

• Convert $8.92 \times 10^{-3}$ to decimal: $0.00892$

• Convert $36,000,000$ to scientific notation: $3.6 \times 10^7$

• Convert $0.00583$ to scientific notation: $5.83 \times 10^{-3}$

Multiplying and Dividing in Scientific Notation

• $(2 \times 10^4)(3 \times 10^6) = 6 \times 10^{4+6} = 6 \times 10^{10}$

• $(5.8 \times 10^7)(6.8 \times 10^{-4}) = (5.8 \times 6.8) \times 10^{7 + (-4)} = 39.44 \times 10^3 = 3.944 \times 10^4$

• $\frac{5.5 \times 10^8}{2.2 \times 10^3} = \frac{5.5}{2.2} \times 10^{8-3} = 2.5 \times 10^5$

• $\frac{8.5 \times 10^4}{3.4 \times 10^{-2}} = 2.5 \times 10^{4 - (-2)} = 2.5 \times 10^6$

Additional info: Some examples and explanations were expanded for clarity and completeness, and some missing steps were logically inferred based on standard college algebra curriculum.