Key Concepts and Problem Types in College Algebra: Equations, Inequalities, Functions, and Applications

Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is raised only to the power of one. Solving these equations involves isolating the variable on one side.

• Definition: A linear equation is an equation of the form ax + b = c, where a, b, and c are constants.

• Example: Solve $5x - 7 = 2x + 8$ Solution: Subtract $2x$ from both sides: $5x - 2x - 7 = 8$ $3x - 7 = 8$ Add $7$to both sides: $3x = 15$ Divide by $3$:$x = 5$

Solving and Expressing Solutions in Interval Notation

Interval notation is a way of writing subsets of the real number line. It is commonly used to express the solution set of inequalities.

• Definition: Interval notation uses parentheses ( ) for open intervals and brackets [ ] for closed intervals.

• Example: Solve and express $-3 < 4x + 1 ext{ } extless ext{ } 9$ in interval notation. Solution: Subtract $1$from all parts:$-4 < 4x < 8$Divide by $4$: $-1 < x < 2$ Interval notation: $( -1, 2 )$

Solving Linear Inequalities

Linear inequalities are similar to linear equations but involve inequality symbols (<, >, ≤, ≥). The solution is often a range of values.

• Example: Solve $2(2x - 5) = 3x - 2 + 4$ Solution: Expand: $4x - 10 = 3x + 2$ Subtract $3x$$:$ Add $10$:$x = 12$

Functions and Their Domains

Domain and Range

The domain of a function is the set of all possible input values (usually x-values), and the range is the set of all possible output values (y-values).

• Definition: For a function $f(x)$, the domain is the set of all $x$ for which $f(x)$ is defined.

• Example: For the relation {(3,3), (1,-8), (3,-1), (8,-3), (1,6)}, the domain is {1, 3, 8} and the range is {3, -8, -1, -3, 6}.

Graphing Linear Equations

Slope and Y-Intercept

Linear equations can be graphed using their slope and y-intercept. The slope indicates the steepness, and the y-intercept is where the line crosses the y-axis.

• Definition: The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Example: Given $y = -2x + 4$, the slope is $-2$ and the y-intercept is $4$.

Word Problems and Applications

Mixture and Value Problems

Word problems often require setting up equations based on the relationships described in the problem. These can include mixture, value, and geometry problems.

• Example: May, Jack, and Brad bought a framed photo for $24.60. May paid $7.20, Jack paid $9.60, and Brad paid the rest. What percent of the total did each pay? Solution: Brad paid $24.60 - (7.20 + 9.60) = $7.80. May:$\frac{7.20}{24.60} \approx 29.3\%$Jack:$\frac{9.60}{24.60} \approx 39.0\%$Brad:$\frac{7.80}{24.60} \approx 31.7\%$

• Example: A rectangle is 3 times as long as it is wide. The perimeter is 88 inches. Find the width. Solution: Let width = $x$, length = $3x$. Perimeter: $2x + 2(3x) = 88$ $2x + 6x = 88$ $8x = 88$ $x = 11$

Systems of Equations

Solving Systems by Substitution or Elimination

Systems of equations involve finding values that satisfy two or more equations simultaneously. Common methods include substitution and elimination.

• Example: Solve the system:

  • $2x + 3y = 8$

  • $x - 4y = 5$

  Solution: Solve one equation for $x$ or $y$ and substitute into the other, or use elimination to combine equations and solve for one variable.

Quadratic Equations

Solving 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 $2x^2 - 5x = 7$ Solution: $2x^2 - 5x - 7 = 0$ Use the quadratic formula with $a=2$, $b=-5$, $c=-7$.

Graphing and Analyzing Functions

Graphing Quadratic Functions

Quadratic functions have the form $y = ax^2 + bx + c$ and their graphs are parabolas. The vertex and axis of symmetry are key features.

• Vertex Formula: $x = -\frac{b}{2a}$

• Example: Graph $y = x^2 - 4x + 3$ Solution: Find vertex, axis of symmetry, and plot points.

Summary Table: Key Problem Types

Problem Type	Key Method	Example
Linear Equation	Isolate variable	$5x - 7 = 2x + 8$
Linear Inequality	Isolate variable, use interval notation	$-3 < 4x + 1 < 9$
System of Equations	Substitution or elimination	$2x + 3y = 8$, $x - 4y = 5$
Quadratic Equation	Factoring, quadratic formula	$2x^2 - 5x = 7$
Word Problem	Translate to equation(s)	Rectangle perimeter, mixture