College Algebra: Solving Equations, Simplifying Expressions, and Solving Inequalities

Solving Linear Equations

Solving for a Variable in Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one. The goal is to isolate the variable on one side of the equation.

• Key Steps:

  1. Combine like terms on each side of the equation.

  2. Use addition or subtraction to move terms containing the variable to one side and constants to the other.

  3. Use multiplication or division to solve for the variable.

• Example: Solve $9x - 5 = 7 + 4x$

  • Subtract $4x$ from both sides: $9x - 4x - 5 = 7$

  • $5x - 5 = 7$

  • Add $5$to both sides: $5x = 12$

  • Divide by $5$:$x = \frac{12}{5}$

Solving Quadratic Equations

Factoring Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. Factoring involves expressing the quadratic as a product of two binomials and setting each factor to zero.

• Key Steps:

  1. Write the equation in standard form: $ax^2 + bx + c = 0$.

  2. Factor the quadratic expression.

  3. Set each factor equal to zero and solve for $x$.

• Example: Solve $x^2 - 9x = 0$

  • Factor: $x(x - 9) = 0$

  • Set each factor to zero: $x = 0$ or $x - 9 = 0$

  • Solutions: $x = 0$, $x = 9$

Square Root Property

The square root property is used when the quadratic equation is in the form $x^2 = d$. The solutions are $x = \pm\sqrt{d}$.

• Example: Solve $6x^2 = 150$

  • Divide both sides by $6$:$x^2 = 25$

  • Take the square root: $x = \pm5$

Completing the Square

Completing the square is a method to solve quadratic equations by converting them into a perfect square trinomial.

• Key Steps:

  1. Move the constant term to the other side.

  2. Add $(\frac{b}{2})^2$ to both sides to complete the square.

  3. Write the left side as a squared binomial and solve for $x$.

• Example: Solve $x^2 + 7x + 4 = 0$

  • Move $4$:$x^2 + 7x = -4$

  • Add $(\frac{7}{2})^2 = 12.25$ to both sides: $x^2 + 7x + 12.25 = 8.25$

  • $(x + 3.5)^2 = 8.25$

  • $x + 3.5 = \pm\sqrt{8.25}$

  • $x = -3.5 \pm \sqrt{8.25}$

Quadratic Formula

The quadratic formula solves any quadratic equation $ax^2 + bx + c = 0$:

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

• Example: Solve $x^2 - 2x + 26 = 0$

  • $a = 1$, $b = -2$, $c = 26$

  • $x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 26}}{2}$

  • $x = \frac{2 \pm \sqrt{4 - 104}}{2}$

  • $x = \frac{2 \pm \sqrt{-100}}{2}$

  • $x = \frac{2 \pm 10i}{2}$

  • $x = 1 \pm 5i$

Simplifying Algebraic and Complex Expressions

Combining Like Terms and Simplifying

To simplify expressions, combine like terms and perform arithmetic operations as indicated.

• Example: $(6 + 5i) - (8 - 6i)$

  • Combine real parts: $6 - 8 = -2$

  • Combine imaginary parts: $5i - (-6i) = 11i$

  • Result: $-2 + 11i$

Multiplying and Dividing Complex Numbers

Complex numbers are in the form $a + bi$, where $i = \sqrt{-1}$. To multiply or divide, use distributive property and rationalize denominators as needed.

• Example (Multiplication): $(4 + 2i)(3 + 8i)$

  • Expand: $4 \times 3 + 4 \times 8i + 2i \times 3 + 2i \times 8i$

  • $= 12 + 32i + 6i + 16i^2$

  • Recall $i^2 = -1$: $16i^2 = -16$

  • Combine: $12 - 16 + 38i = -4 + 38i$

• Example (Division): $\frac{9}{4 - 2i}$

  • Multiply numerator and denominator by the conjugate $(4 + 2i)$:

  • $\frac{9(4 + 2i)}{(4 - 2i)(4 + 2i)}$

  • Denominator: $16 + 8i - 8i - 4i^2 = 16 + 4 = 20$

  • Numerator: $36 + 18i$

  • Result: $\frac{36 + 18i}{20} = \frac{18 + 9i}{10}$

Solving Equations with Rational Exponents

Equations Involving Rational Exponents

Rational exponents represent roots and powers. For example, $x^{m/n} = \sqrt[n]{x^m}$.

• Example: $(x + 35)^{3/2} = 343$

  • Take both sides to the $2/3$ power: $x + 35 = 343^{2/3}$

  • $343^{1/3} = 7$, so $343^{2/3} = 7^2 = 49$

  • $x + 35 = 49$

  • $x = 14$

Solving Inequalities and Interval Notation

Solving Linear Inequalities

Linear inequalities are solved similarly to linear equations, but the solution is a range of values. When multiplying or dividing both sides by a negative number, reverse the inequality sign.

• Key Steps:

  1. Isolate the variable on one side.

  2. Solve for the variable.

  3. Express the solution in interval notation.

• Example: $6x + 5 \leq 4x + 19$

  • Subtract $4x$: $2x + 5 \leq 19$

  • Subtract $5$: $2x \leq 14$

  • Divide by $2$:$x \leq 7$

  • Interval notation: $(-\infty, 7]$

Graphing Solutions on a Number Line

Solutions to inequalities can be represented on a number line, with open or closed circles indicating whether endpoints are included.

• Example: $-6 < x < 7$

  • Interval notation: $(-6, 7)$

  • Graph: Open circles at $-6$ and $7$, shade between.

Compound Inequalities

Compound inequalities involve two inequalities joined by "and" or "or". The solution is the intersection (for "and") or union (for "or") of the individual solutions.

• Example: $6x - 5 \geq 8$ or $6x - 5 \leq -8$

  • First: $6x - 5 \geq 8 \implies x \geq \frac{13}{6}$

  • Second: $6x - 5 \leq -8 \implies x \leq -\frac{1}{2}$

  • Interval notation: $(-\infty, -\frac{1}{2}] \cup [\frac{13}{6}, \infty)$

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Example
Factoring	When the quadratic can be factored easily	$x^2 - 9x = 0$
Square Root Property	When the equation is in the form $x^2 = d$	$x^2 = 25$
Completing the Square	When $a = 1$ and $b$ is even, or to derive the quadratic formula	$x^2 + 7x + 4 = 0$
Quadratic Formula	Any quadratic equation	$x^2 - 2x + 26 = 0$

Additional info: Some steps and explanations have been expanded for clarity and completeness, as the original handwritten notes were brief and sometimes omitted intermediate steps.