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 $3(7x - 5) - (6x - 9) = 5x + 12$

  • Expand: $21x - 15 - 6x + 9 = 5x + 12$

  • Combine like terms: $15x - 6 = 5x + 12$

  • Subtract $5x$: $10x - 6 = 12$

  • Add $6$: $10x = 18$

  • Divide by $10$:$x = \frac{9}{5}$

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.

• 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: $(x-9)(x+9)=0$

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

  • Solutions: $x=9$ or $x=-9$

Square Root Property

The square root property is used when the equation is in the form $x^2 = a$.

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

• 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 quadratic equations by rewriting them 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: $x + 3.5 = \pm\sqrt{8.25}$

  • Solve for $x$

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

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

Simplifying Expressions

Combining Like Terms and Complex Numbers

When simplifying, 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: $-2 + 11i$

• Example: $\sqrt{-8} + \sqrt{-17}$

  • $\sqrt{-8} = 2\sqrt{2}i$

  • $\sqrt{-17} = \sqrt{17}i$

  • Result: $2\sqrt{2}i + \sqrt{17}i$

Solving Inequalities

Linear Inequalities

Linear inequalities are solved similarly to linear equations, but the solution is often an interval rather than a single value.

• Key Steps:

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

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

  3. Express the solution in interval notation and graph it on a number line.

• 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: $(-\infty, 7]$

Compound Inequalities

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

• Example: $-5 \leq 3x + 13 \leq 34$

  • Subtract $13$:$-18 \leq 3x \leq 21$

  • Divide by $3$:$-6 \leq x \leq 7$

  • Interval notation: $[-6, 7]$

Graphing Solutions

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

• Closed circle: Endpoint is included (≤ or ≥).

• Open circle: Endpoint is not included (< or >).

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Example
Factoring	When the quadratic can be factored easily	$(x-3)(x+2)=0$
Square Root Property	When the equation is in the form $x^2 = a$	$x^2 = 25$
Completing the Square	When $a=1$ and $b$ is even, or to derive the quadratic formula	$x^2 + 6x + 5 = 0$
Quadratic Formula	Any quadratic equation	$x^2 - 2x + 26 = 0$

Additional info:

• Some problems involve complex numbers, where $i = \sqrt{-1}$.

• Interval notation is used to express solution sets for inequalities.

• All examples and methods are standard in College Algebra curricula.