College Algebra Study Guide: Equations, Inequalities, and Complex Numbers

Graphs of Linear Equations

General and Slope-Intercept Form

Linear equations can be represented in different forms, each providing unique insights into the relationship between variables.

• General Form: $Ax + By = C$

• Slope-Intercept Form: $y = mx + b$, where m is the slope and b is the y-intercept.

• Example: $y = -\frac{3}{4}x + 12$

• Example: $y = -\frac{1}{2}x + 4$

Graphing these equations helps visualize the relationship between x and y.

Linear and Rational Equations

Solving Linear Equations with Fractions

To solve linear equations involving fractions, use the following method:

• Multiply both sides by the least common denominator (LCD) to clear fractions.

• Check for extraneous solutions (solutions that do not satisfy the original equation).

• Example: $\frac{1}{2}x - \frac{2}{3} = \frac{5}{3}$

• Example: $\frac{2}{1}x - \frac{1}{5} = \frac{6}{4}x + \frac{1}{6}$

Types of Equations

• Conditional: True for some values.

• Identity: True for all values.

• Contradiction: True for no values.

• Examples:

  • $x + 5 = 2x - 7$ (Conditional)

  • $3(x - 4) - 2x = x - 12$ (Identity or Conditional)

  • $x = x + 1$ (Contradiction)

Mathematical Models and Story Problems

Strategy for Solving Story Problems

Solving real-world problems requires translating the scenario into mathematical language.

1. Analyze the problem and define variables.

2. Set up an equation based on the relationships described.

3. Solve the equation.

4. Check (verify the solution in context).

5. State the solution with appropriate units.

Additional info: Always ensure the answer makes sense in the context of the problem.

Complex Numbers

Simplifying Imaginary Numbers

Imaginary numbers arise when taking the square root of negative numbers.

• Example: $\sqrt{-36} = 6i$

• Example: $\sqrt{-200} = 10i\sqrt{2}$

Every number is a complex number, which includes both real and imaginary parts.

• Complex numbers are all the real and imaginary numbers together.

• Every complex number can be written in the form $a + bi$.

Operations with Complex Numbers

• Addition: $(3 + 4i) + (1 - 5i) = 4 - i$

• Subtraction: $(3 + 4i) - (1 - 5i) = 2 + 9i$

• Multiplication: $(3 + 4i)(1 - 5i)$

• Division: $\frac{3 + 4i}{1 - 5i}$

Absolute Value: $|a + bi| = \sqrt{a^2 + b^2}$

• Example: $|3 + 4i| = \sqrt{3^2 + 4^2} = 5$

Additional info: Always convert to "i" before performing operations.

Quadratic Equations

Techniques for Solving Quadratic Equations

There are four main techniques for solving quadratic equations:

1. Factoring: Set the equation to zero and factor.

   • Example: $x^2 - 4x - 15 = 0$

   • Example: $5x^2 = 81$

2. Square Root Property: If $x^2 = k$, then $x = \pm \sqrt{k}$.

   • Example: $x^2 - 5 = 20 \Rightarrow x^2 = 25 \Rightarrow x = \pm 5$

   • Example: $a^2 - 7 = 150 - 200$

3. Completing the Square: Works for any quadratic.

   1. Move the constant to the other side.

   2. Factor out or divide by "a" if necessary.

   3. Take half the coefficient of x, square it, and add to both sides.

   4. Rewrite as a perfect square.

   5. Finish using the square root property.

   • Example: $x^2 + 10x + 1 = 0$

   • Example: $x^2 + 5x - 45 = 35$

4. Quadratic Formula: Solves any quadratic equation.

   • Equation must be set to zero.

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

   • Example: $x^2 + 2x - 9 = 0$

   • Example: $2x^2 + x - 1 = 1$

Other Equations

Equations in Quadratic Form

Some equations are not quadratics but can be solved using quadratic techniques.

• Example: $x^4 - 2x^2 - 12 = 0$

• Example: $x^{2/3} - 3x^{1/3} - 4 = 0$

Solving Equations with Roots

1. Isolate a radical.

2. Raise both sides to the appropriate power (repeat if necessary).

3. Solve the resulting equation.

4. Check for extraneous solutions.

• Example: $\sqrt{x + 5} = x - 1$

• Example: $\sqrt{3x} = x$

• Example: $\sqrt{4y + 1} = 6 - y$

Solving Equations with Rational Exponents

Isolate the variable with the rational exponent and raise both sides to the reciprocal power.

• Example: $x^{2/3} = 5$

• Example: $(x + 7)^{2/5} - 3 = 13$

Solving Equations with Absolute Values

1. Isolate the absolute value.

2. Separate into two equations:

   • One equation as written.

   • Other equation with the "answer" negated.

3. Solve both equations.

4. Check solutions.

• Example: $|4x - 5| = 6$

• Example: $|3x - 2| + 1 = 10$

• Example: $|x^2 + 2x| = x + 7$

Linear Inequalities and Absolute Value Inequalities

Solving Linear Inequalities

Linear inequalities are solved similarly to equations, but require careful attention to the direction of the inequality.

• Example: $3x - 7 < 8$

• Example: $2(1 - x) < -12$

Use the "S.P.I.T." method: Set, Plot, Interval Test

1. Rewrite as equation(s) and solve.

2. Plot these numbers on a number line.

3. Test values from the intervals to see if they satisfy the inequality.

• Example: $7y - 5 > 23 - y$

Compound Inequalities

Compound inequalities can be solved using the S.P.I.T. method, avoiding confusion with "and/or" logic.

• Example: $7 < 2x - 1 < 11$

• Example: $3x + 4 < -2$ or $3x + 4 \geq 10$

Absolute Value Inequalities

Combine absolute value with inequalities using the S.P.I.T. method for clarity.

• Example: $|3x - 1| < 8$

• Example: $|5x + 3| \geq 6 - 9x$

Additional info: Solutions should be given in graphs and interval notation for clarity.