Elementary Algebra

 

1. Introduction to Algebra

• Definition: Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. The symbols (often letters) represent numbers and quantities in formulas and equations.
• Expressions vs. Equations:
• Expression: A combination of numbers, variables, and operations (e.g., $3x+\mathrm{<span\; class="base"><span\; class="strut"></span><span\; class="mord">5</span>).</span>}$
• Equation: A statement that two expressions are equal (e.g., $3x+5=\mathrm{<span\; class="base"><span\; class="strut"></span><span\; class="mord">11</span>).</span>}$

2. Basic Concepts

• Variables: Symbols (typically letters) that represent unknown values. Common examples include $x$, $y$, and $z$.
• Constants: Fixed values, like numbers (e.g., 2, -3, 4.5).
• Coefficients: Numbers that multiply a variable (e.g., in $3x$, 3 is the coefficient).
• Terms: Individual parts of an expression, separated by addition or subtraction (e.g., in $3x + 5$
  $3x$ and 5 are terms).
• Polynomials: Expressions that consist of multiple terms (e.g., $2x^2+3x+\mathrm{4).}$

3. Operations on Algebraic Expressions

• Addition/Subtraction: Combine like terms (e.g., 3x+2x=5x).
• Multiplication: Use distributive property (e.g., $2(x + 3) = 2x + 6$).
• Division: Divide coefficients and subtract exponents for similar bases (e.g., $\frac{6x^2}{3x} = 2x$).

4. Solving Linear Equations

• Linear Equation: An equation of the first degree (e.g., $ax+b=\mathrm{<span\; class="base"><span\; class="strut"></span><span\; class="mord">0</span>).</span>}$
• Solution Steps:
  1. Simplify both sides (if needed).
  2. Isolate the variable (e.g., move terms involving $x$ to one side and constants to the other).
  3. Solve for the variable (e.g., $x = \frac{-b}{a}$).
• Examples:
• $2x + 3 = 7$ → $2x = 4$→ $x = 2$
  
• $3x - 4 = 2x + 1$ → $x = 5$
  

5. Inequalities

• Definition: Similar to equations but involve inequalities ($<$, $\leq$, $>$, $\geq$).
• Solving Steps:
  1. Simplify both sides.
  2. Isolate the variable.
  3. Solve, remembering to reverse the inequality sign when multiplying or dividing by a negative number.
• Graphical Representation: Inequalities can be represented on a number line.

6. Factoring

• Definition: Breaking down a complex expression into a product of simpler expressions.
• Common Methods:
  • Common Factor: Factor out the greatest common factor (GCF) (e.g., $6x + 9 = 3(2x + 3)$
    
  • Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$
    
  • Quadratic Trinomials: $a{x}^2+bx+\mathrm{c\; can\; sometimes\; be\; factored\; into }$$(mx + n)(px + q)$
    
• Examples:
• $x^2 - 9 = (x + 3)(x - 3)$
  
• $x^2 + 5x + 6 = (x + 2)(x + 3)$
  

7. Quadratic Equations

• Definition: Equations of the form $ax^2 + bx + c = 0$.
• Solution Methods:
  • Factoring: (if factorable) $x^2 + 5x + 6 = (x + 2)(x + 3) = 0$
  • $so,\; x=-2$ or x = −3
  • Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
  • Completing the Square: Transform the equation into a perfect square trinomial and solve.

8. Exponents and Powers

• Rules:
  • $a^m \times a^n = a^{m+n}$
    
  • $\frac{a^m}{a^n}=a^{m-n}$
  • $(a^m{)}^n=a^{mn}$
  • $a^{-n} = \frac{1}{a^n}$
    
  • $a^0 = 1$ (where $a\mathrm{\ne }\mathrm{0).}$
• Examples:
• $2^3\times 2^2=2^5=$.
• $\frac{x^5}{x^2} = x^{3}$
  

9. Radicals and Rational Exponents

• Radicals: Expressions involving roots, such as $\sqrt{x}$ (square root).
• Properties:
  • $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$.
  • $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
• Rational Exponents: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$
  
  • Example: $x^{\frac{1}{2}} = \sqrt{x}$
    

10. Functions

• Definition: A relationship between two sets where each input (x) has exactly one output (y).
• Notation: $f(x),\; where$$f$ is the function, and $x$ is the input.
• Linear Function: $f(x) = mx + b$ (straight-line graph).
• Quadratic Function: $f(x) = ax^2 + bx + c$ (parabola-shaped graph).

11. Systems of Equations

• Definition: A set of two or more equations with the same variables.
• Solution Methods:
  • Substitution: Solve one equation for one variable and substitute into the other.
  • Elimination: Add or subtract equations to eliminate one variable.
  • Graphical Method: Graph both equations and find the point(s) of the intersection.
• Example:
  • $x+y=\mathrm{3\; and }$$2x - y = 1$
    

• Steps to Solve:
  1. Define the variables.
  2. Write an equation based on the problem.
  3. Solve the equation.
  4. Interpret the solution in the context of the problem.
• Examples:
  • If a number is doubled and then increased by 5, the result is 15. What is the number? (Equation: $2x + 5 = 15$).

13. Rational Expressions and Equations

• Rational Expressions: Fractions where the numerator and/or denominator is a polynomial (e.g., $\frac{2x + 3}{x - 1}$).
• Operations:
  • Addition/Subtraction: Find a common denominator.
  • Multiplication: Multiply numerators together and denominators together.
  • Division: Multiply by the reciprocal of the divisor.
• Solving Rational Equations: Clear fractions by multiplying by the least common denominator (LCD), then solve the resulting equation.

14. Absolute Value

• Definition: The distance of a number from zero on a number line, is always non-negative.
  • $|x| = x$ if $x \geq 0$.
  • $|x| = -x$ if $x < 0$.
• Absolute Value Equations: Solve by considering both the positive and negative cases.
  • Example: $|x - 3| = 5$ leads to
  •   $x - 3 = 5$ or $x - 3 = -5$, 
  • so $x = 8$ or $x = -2$
    

15. Conclusion

Elementary Algebra forms the foundation for more advanced mathematical concepts. These basic principles are crucial for success in higher-level mathematics, including geometry, calculus, and beyond.