College Algebra: Solving Equations, Simplifying Expressions, and Interval Notation

Solving Linear Equations

Single Variable Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one. Solving these equations involves isolating the variable on one side of the equation.

• Key Steps:

  1. Expand expressions and combine like terms.

  2. Move all terms containing the variable to one side.

  3. Isolate the variable by performing inverse operations.

• Example: Solve for $x$: $5(x + 3) + 4x - 3 = 2(x - 4)$ Solution:

  1. Expand: $5x + 15 + 4x - 3 = 2x - 8$

  2. Combine like terms: $9x + 12 = 2x - 8$

  3. Subtract $2x$ from both sides: $7x + 12 = -8$

  4. Subtract $12$: $7x = -20$

  5. Divide by $7$:$x = -\frac{20}{7}$

Solving Quadratic Equations

Standard Form and Factoring

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: Solve for $x$: $x^2 - 2x - 10 = 0$ Solution:

  1. Identify $a = 1$, $b = -2$, $c = -10$

  2. Apply formula: $x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-10)}}{2}$

  3. $x = \frac{2 \pm \sqrt{4 + 40}}{2} = \frac{2 \pm \sqrt{44}}{2}$

  4. $x = \frac{2 \pm 2\sqrt{11}}{2} = 1 \pm \sqrt{11}$

Solving Rational Equations

Equations Involving Fractions

Rational equations contain fractions with polynomials in the numerator and denominator. To solve, find a common denominator and clear fractions.

• Example: Solve for $x$: $\frac{3x+2}{x^2} + 1 = \frac{-2}{7-2x}$ Solution:

  1. Find common denominator and multiply both sides to eliminate fractions.

  2. Solve resulting polynomial equation.

Simplifying Algebraic Expressions

Combining Like Terms and Reducing Fractions

Simplifying expressions involves combining like terms and reducing fractions to their simplest form.

• Example: Simplify $\frac{1}{x+2} - \frac{1}{x^2+2x}$ Solution:

  1. Factor denominator: $x^2 + 2x = x(x+2)$

  2. Rewrite: $\frac{1}{x+2} - \frac{1}{x(x+2)}$

  3. Common denominator: $x(x+2)$

  4. $\frac{x}{x(x+2)} - \frac{1}{x(x+2)} = \frac{x-1}{x(x+2)}$

Word Problems: Perimeter and Area

Application of Algebra to Geometry

Algebra can be used to solve geometric problems involving perimeter and area. Translate the word problem into equations and solve for the unknown.

• Example: If the length of each side of a square is increased by 3 cm, the perimeter of the new square is 40 cm more than twice the length of the original square. Find the dimensions of the original square. Solution:

  1. Let $x$ = original side length.

  2. New side length = $x + 3$

  3. Perimeter of new square = $4(x + 3)$

  4. Twice original length = $2x$

  5. Set up equation: $4(x + 3) = 2x + 40$

  6. Solve: $4x + 12 = 2x + 40 \Rightarrow 2x = 28 \Rightarrow x = 14$

Radical Equations

Solving Equations with Square Roots

Radical equations contain variables inside a root. Isolate the radical and square both sides to eliminate the root.

• Example: Solve for $x$: $x = \sqrt{x + 12}$ Solution:

  1. Square both sides: $x^2 = x + 12$

  2. Rearrange: $x^2 - x - 12 = 0$

  3. Factor: $(x - 4)(x + 3) = 0$

  4. Solutions: $x = 4$, $x = -3$ (check for extraneous solutions)

Interval Notation

Expressing Solution Sets

Interval notation is a way to describe sets of numbers, often solutions to inequalities. Use parentheses for open intervals and brackets for closed intervals.

• Example: Solve $x^2 - 2x - 15 \leq 0$ and write the answer in interval notation. Solution:

  1. Factor: $(x - 5)(x + 3) \leq 0$

  2. Find zeros: $x = 5$, $x = -3$

  3. Test intervals: Solution is $[-3, 5]$

Summary Table: Types of Equations and Solution Methods

Type of Equation	General Form	Solution Method
Linear	$ax + b = c$	Isolate variable, inverse operations
Quadratic	$ax^2 + bx + c = 0$	Factoring, quadratic formula, completing the square
Rational	$\frac{P(x)}{Q(x)} = R(x)$	Find common denominator, clear fractions
Radical	$\sqrt{x + a} = b$	Isolate radical, square both sides
Inequality	$ax^2 + bx + c \leq 0$	Find zeros, test intervals, write in interval notation

Additional info:

• Some questions involve interval notation and require understanding of set notation.

• Problems cover a range of College Algebra topics: linear, quadratic, rational, radical equations, and word problems.