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 variable terms to one side and constants to the other.

  3. Divide or multiply 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 $5$:$x = \frac{12}{5}$

Simplifying Algebraic Expressions

Combining Like Terms and Simplifying Complex Numbers

Algebraic expressions can often be simplified by combining like terms and applying arithmetic operations. When working with complex numbers, use the property $i^2 = -1$.

• Key Points:

  • Combine real parts and imaginary parts separately.

  • Apply the distributive property as needed.

• Example: $(6 + 5i) - (8 - 6i)$

  • Combine real parts: $6 - 8 = -2$

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

  • Result: $-2 + 11i$

• Example: $(-4 + 2i) + (4 + 8i)$

  • Combine real parts: $-4 + 4 = 0$

  • Combine imaginary parts: $2i + 8i = 10i$

  • Result: $10i$

Solving Quadratic Equations

Factoring Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. Factoring is one method to solve them by expressing the quadratic as a product of two binomials.

• Key Steps:

  1. Set the equation to zero.

  2. Factor the quadratic expression.

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

• Example: $x^2 - 9x + 8 = 0$

  • Factor: $(x - 8)(x - 1) = 0$

  • Set each factor to zero: $x - 8 = 0$ or $x - 1 = 0$

  • Solutions: $x = 8$ or $x = 1$

Square Root Property

The square root property is used when the quadratic equation is in the form $(x - h)^2 = k$.

• Formula: $(x - h)^2 = k \implies x - h = \pm\sqrt{k}$

• Example: $(x - 2)^2 = 96$

  • Take square root: $x - 2 = \pm\sqrt{96}$

  • Simplify: $x = 2 \pm 4\sqrt{6}$

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 using the square root property.

• Example: $x^2 + 7x + 4 = 0$

  • Move $4$:$x^2 + 7x = -4$

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

  • Rewrite: $(x + 3.5)^2 = 8.25$

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

  • Solutions: $x = -3.5 \pm \sqrt{8.25}$

Quadratic Formula

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

• Formula:

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

• Example: $x^2 - 2x + 26 = 0$

  • $a = 1$, $b = -2$, $c = 26$

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

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

Solving Radical Equations

Equations Involving Square Roots

Radical equations contain variables inside a radical, usually a square root. To solve, isolate the radical and square both sides.

• Key Steps:

  1. Isolate the radical on one side.

  2. Square both sides to eliminate the radical.

  3. Solve the resulting equation.

  4. Check for extraneous solutions.

• Example: $\sqrt{3} (\sqrt{3} - \sqrt{17})$

  • Apply distributive property: $3 - \sqrt{51}$

Solving Inequalities

Linear Inequalities and Interval Notation

Linear inequalities are similar to linear equations but use inequality symbols (<, >, ≤, ≥). Solutions are often expressed in interval notation and can be graphed on a number line.

• Key Steps:

  1. Solve the inequality as you would an equation.

  2. If you multiply or divide by a negative number, reverse the inequality sign.

  3. Express the solution in interval notation.

  4. Graph the solution on a number line.

• Example: $6x - 5 \leq 8$ or $6x - 5 \geq 13$

  • First inequality: $6x \leq 13 \implies x \leq \frac{13}{6}$

  • Second inequality: $6x \geq 18 \implies x \geq 3$

  • Solution: $(-\infty, \frac{13}{6}] \cup [3, \infty)$

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Example
Factoring	When the quadratic factors easily	$x^2 - 9x + 8 = 0$
Square Root Property	When the equation is in the form $(x - h)^2 = k$	$(x - 2)^2 = 96$
Completing the Square	When $a = 1$ and $b$ is even, or to derive the quadratic formula	$x^2 + 7x + 4 = 0$
Quadratic Formula	Always works for any quadratic equation	$x^2 - 2x + 26 = 0$

Additional info:

• Some problems involve complex numbers, which are numbers of the form $a + bi$ where $i = \sqrt{-1}$.

• Interval notation is used to describe sets of solutions for inequalities, e.g., $(-\infty, 7]$.

• Always check for extraneous solutions when solving radical equations.