Study Notes: Equations and Inequalities in College Algebra

I Equations and Inequalities

Linear Equations and Solutions

Linear equations are fundamental in algebra and involve variables raised only to the first power. Solving these equations is essential for understanding more complex algebraic concepts.

• Definition: A linear equation in one variable has the form $ax + b = 0$, where $a$ and $b$ are constants.

• Solution: To solve for $x$, isolate the variable:

$x = -\frac{b}{a}$

• Example: Solve $6 - x = 4$

$6 - x = 4$ $-x = 4 - 6$ $-x = -2$ $x = 2$

Systems of Linear Equations

Systems of equations involve finding values that satisfy multiple equations simultaneously. These are often solved using substitution, elimination, or matrix methods.

• Definition: A system of linear equations consists of two or more equations with the same variables.

• Example: Solve the system:

  • $x + y = 2$

  • $x - y = 4$

Add the equations: $(x + y) + (x - y) = 2 + 4$ $2x = 6$ $x = 3$ Substitute $x$ into the first equation: $3 + y = 2$ $y = -1$

Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula.

• Quadratic Formula:

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

• Example: Solve $x^2 - 4x + 3 = 0$

Factor: $(x - 1)(x - 3) = 0$ Solutions: $x = 1$, $x = 3$

Absolute Value Equations

Absolute value equations involve expressions within absolute value bars. The solution often requires considering two cases.

• Definition: $|x| = a$ implies $x = a$ or $x = -a$.

• Example: Solve $|x - 2| = 5$

$x - 2 = 5$ or $x - 2 = -5$ $x = 7$ or $x = -3$

Linear Inequalities

Linear inequalities are similar to linear equations but involve inequality signs. The solution is often a range of values.

• Definition: An inequality such as $ax + b > 0$ or $ax + b < 0$.

• Solving: Isolate the variable and consider the direction of the inequality.

• Example: Solve $2x - 3 < 7$

$2x - 3 < 7$ $2x < 10$ $x < 5$

Systems of Inequalities

Systems of inequalities involve finding the set of values that satisfy all inequalities in the system. The solution is often represented graphically.

• Example: Solve the system:

  • $x + y > 1$

  • $x - y < 3$

The solution is the region in the plane where both inequalities are true.

Special Cases and Notation

Some equations and inequalities involve special notation or cases, such as piecewise functions or absolute value expressions.

• Piecewise Functions: Functions defined by different expressions depending on the input value.

• Example: $f(x) = \begin{cases} x + 2 & x \geq 0 \\ -x & x < 0 \end{cases}$

$f(3) = 3 + 2 = 5$ $f(-2) = -(-2) = 2$

Summary Table: Common Equation Types

The following table summarizes common types of equations and their solution methods.

Type	General Form	Solution Method	Example
Linear	$ax + b = 0$	Isolate $x$	$2x + 3 = 7$
Quadratic	$ax^2 + bx + c = 0$	Factoring, Quadratic Formula	$x^2 - 4x + 3 = 0$
Absolute Value	$|x| = a$	Two cases: $x = a$ or $x = -a$	$|x - 2| = 5$
System	Multiple equations	Substitution, Elimination	$x + y = 2$, $x - y = 4$
Inequality	$ax + b > 0$	Isolate $x$, graph solution	$2x - 3 < 7$

Additional info: Some content was inferred from context and standard college algebra topics, as the original notes were fragmented and partially unclear.