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

Solving Linear Equations

Solving for a Variable

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: $9x - 4x - 5 = 7$

  • Add $5$to both sides: $5x = 12$

  • Divide by $5$:$x = \frac{12}{5}$

Solving Equations with Parentheses and Distribution

Using the Distributive Property

When equations contain parentheses, use the distributive property to eliminate them before solving.

• Distributive Property: $a(b + c) = ab + ac$

• Example: Solve $3(7x - 5) - 6(x - 9) = 5x + 12$

  • Expand: $21x - 15 - 6x + 54 = 5x + 12$

  • Combine like terms: $15x + 39 = 5x + 12$

  • Subtract $5x$: $10x + 39 = 12$

  • Subtract $39$: $10x = -27$

  • Divide by $10$:$x = -\frac{27}{10}$

Solving Quadratic Equations

Factoring Method

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. Factoring is one method to solve them.

• Key Steps:

  1. Set the equation to zero.

  2. Factor the quadratic expression.

  3. Set each factor equal to zero and solve for the variable.

• Example: $x^2 - 9x + 20 = 0$

  • Factor: $(x - 4)(x - 5) = 0$

  • Set each factor to zero: $x - 4 = 0$ or $x - 5 = 0$

  • Solutions: $x = 4$ or $x = 5$

Square Root Property

For equations of the form $x^2 = k$, take the square root of both sides.

• Formula: $x^2 = k \implies x = \pm\sqrt{k}$

• Example: $6x^2 = 150$

  • Divide by $6$:$x^2 = 25$

  • Take square root: $x = \pm5$

Completing the Square

Completing the square is a method to solve any quadratic equation by rewriting it in the form $(x + p)^2 = q$.

• 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. Rewrite as a squared binomial and solve for $x$.

• Example: $x^2 + 7x + 4 = 0$

  • Move $4$:$x^2 + 7x = -4$

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

  • Rewrite: $(x + 3.5)^2 = 8.25$

  • Take square root: $x + 3.5 = \pm\sqrt{8.25}$

  • Solve for $x$

Quadratic Formula

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

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

• Example: $x^2 - 2x + 26 = 0$

  • $a = 1$, $b = -2$, $c = 26$

  • Discriminant: $(-2)^2 - 4(1)(26) = 4 - 104 = -100$

  • $x = \frac{2 \pm \sqrt{-100}}{2} = \frac{2 \pm 10i}{2} = 1 \pm 5i$

Simplifying Expressions

Combining Like Terms and Complex Numbers

To simplify expressions, combine like terms and use the properties of imaginary numbers ($i^2 = -1$).

• Example: $(6 + 5i) - (8 - 6i)$

  • Combine real parts: $6 - 8 = -2$

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

  • Result: $-2 + 11i$

• Multiplying Complex Numbers:

  • $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

  • Example: $(-4 + 2i)(4 + 3i)$

    • Expand: $-16 - 12i + 8i + 6i^2$

    • Recall $i^2 = -1$: $6i^2 = -6$

    • Combine: $-16 - 6 = -22$ (real), $-12i + 8i = -4i$

    • Result: $-22 - 4i$

Solving Radical Equations

Isolating and Squaring Both Sides

To solve equations with square roots, isolate the radical and square both sides to eliminate it.

• Example: $\sqrt{8 - x} = \sqrt{3x - 17}$

  • Square both sides: $8 - x = 3x - 17$

  • Solve for $x$: $8 + 17 = 3x + x \implies 25 = 4x \implies x = 6.25$

Solving Inequalities

Linear Inequalities

Solving inequalities is similar to solving equations, but the solution is often a range of values. When multiplying or dividing both sides by a negative number, reverse the inequality sign.

• Key Steps:

  1. Isolate the variable using addition, subtraction, multiplication, or division.

  2. Express the solution in interval notation.

  3. Graph the solution on a number line.

• Example: $6x - 5 \geq 8$

  • Add $5$: $6x \geq 13$

  • Divide by $6$:$x \geq \frac{13}{6}$

  • Interval notation: $[\frac{13}{6}, \infty)$

Compound Inequalities

Compound inequalities involve two inequalities joined by 'and' or 'or'.

• Example: $-5 \leq 3x + 3 \leq 34$

  • Subtract $3$:$-8 \leq 3x \leq 31$

  • Divide by $3$:$-\frac{8}{3} \leq x \leq \frac{31}{3}$

  • Interval notation: $[-\frac{8}{3}, \frac{31}{3}]$

Interval Notation and Graphing Solutions

Interval Notation

Interval notation is a way to describe sets of numbers between two endpoints.

• Types:

  • Open interval: $(a, b)$ (does not include endpoints)

  • Closed interval: $[a, b]$ (includes endpoints)

  • Half-open interval: $[a, b)$ or $(a, b]$

  • Unbounded intervals: $(-\infty, a)$, $(a, \infty)$, etc.

• Example: $x > 2$ is $(2, \infty)$

Summary Table: Methods for Solving Equations

Equation Type	Method	Example
Linear	Isolate variable	$2x + 3 = 7$
Quadratic (factorable)	Factoring	$x^2 - 5x + 6 = 0$
Quadratic (not factorable)	Quadratic formula	$x^2 + x + 1 = 0$
Quadratic (perfect square)	Square root property	$x^2 = 9$
Radical	Isolate radical, square both sides	$\sqrt{x + 1} = 5$
Inequality	Isolate variable, use interval notation	$3x - 2 > 4$

Additional info: The above notes are based on handwritten solutions to typical College Algebra problems, including linear and quadratic equations, complex numbers, radical equations, and inequalities. Interval notation and graphing are also included as essential skills for expressing solution sets.