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. 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 $5$:$x = \frac{12}{5}$

Simplifying Algebraic Expressions

Combining Like Terms and Operations with Complex Numbers

Algebraic expressions can include real and imaginary numbers. Simplifying involves combining like terms and applying arithmetic operations.

• Key Points:

  • Like terms have the same variable raised to the same power.

  • For complex numbers, $i$ is the imaginary unit where $i^2 = -1$.

• Example: $(6 + 5i) - (8 - 6i)$

  • Distribute the negative: $6 + 5i - 8 + 6i$

  • Combine real parts: $6 - 8 = -2$

  • Combine imaginary parts: $5i + 6i = 11i$

  • Result: $-2 + 11i$

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.

• 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: $x^2 - 9x = 0$

  • Factor: $x(x - 9) = 0$

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

Square Root Property

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

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

• Example: $x^2 = 25$

  • $x = \pm\sqrt{25}$

  • $x = 5$ or $x = -5$

Completing the Square

Completing the square is a method to solve any quadratic equation by converting it 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. 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: $x^2 + 7x + 12.25 = 8.25$

  • $(x + 3.5)^2 = 8.25$

  • $x + 3.5 = \pm\sqrt{8.25}$

  • $x = -3.5 \pm \sqrt{8.25}$

Quadratic Formula

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

• Formula:

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

• Example: $x^2 - 2x + 26 = 0$

  • $a = 1$, $b = -2$, $c = 26$

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

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

  • $x = \frac{2 \pm \sqrt{-100}}{2}$

  • $x = \frac{2 \pm 10i}{2}$

  • $x = 1 \pm 5i$

Simplifying Radical Expressions

Operations with Radicals

Radical expressions involve roots, such as square roots or cube roots. Simplifying involves combining like radicals and rationalizing denominators if necessary.

• Example: $\sqrt{8} - \sqrt{18} + \sqrt{50}$

  • $\sqrt{8} = 2\sqrt{2}$, $\sqrt{18} = 3\sqrt{2}$, $\sqrt{50} = 5\sqrt{2}$

  • Combine: $2\sqrt{2} - 3\sqrt{2} + 5\sqrt{2} = 4\sqrt{2}$

Solving Inequalities

Linear Inequalities

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

• Key Steps:

  1. Isolate the variable on one side.

  2. Express the solution in interval notation.

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

Compound Inequalities

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

• Example: $-5 < 3x + 13 \leq 34$

  • Subtract $13$:$-18 < 3x \leq 21$

  • Divide by $3$:$-6 < x \leq 7$

  • Interval notation: $(-6, 7]$

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Example
Factoring	When the quadratic factors easily	$x^2 - 9x = 0$
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	$x^2 - 2x + 26 = 0$