Equations and Inequalities

Absolute Value Equations

Absolute value equations require splitting into two cases to solve for the variable.

• Definition: The absolute value of a number is its distance from zero on the number line, always non-negative.

• Solving |ax + b| = c: Set up two equations: ax + b = c and ax + b = -c, then solve each for x.

• Example: Solve .

  • Case 1:

  • Case 2:

  • Solution: or

Completing the Square

This method rewrites a quadratic equation in the form of a perfect square trinomial to solve for x.

• Process: For , add to both sides to complete the square.

• Example: Solve by completing the square.

  • Add to both sides:

  • So,

  • Take square roots:

  • Solutions: or

Quadratic Inequalities

Quadratic inequalities are solved by finding critical values and testing intervals.

• Steps:

  1. Set the quadratic equal to zero to find critical values.

  2. Test intervals between and beyond these values to determine where the inequality holds.

• Example: Solve .

  • Rewrite:

  • Factor:

  • Critical values: ,

  • Test intervals:

  • Solution: or

Complex Numbers

Complex numbers are numbers in the form , where .

• Definition: is the imaginary unit, satisfying .

• Example: Solve .

  • Solution: or

Functions and Their Graphs

Domain and Range

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

• Domain: Exclude values that make denominators zero or radicands of even roots negative.

• Example: Find the domain of .

  • Set

  • Domain:

Analyzing Graphs: Increasing/Decreasing Intervals

Determine where a function is increasing or decreasing by observing the graph from left to right.

• Increasing: The function rises as x increases.

• Decreasing: The function falls as x increases.

• Intervals are written using x-values.

Distance Formula

The distance between two points and is given by:

• Example: Find the distance between (1, 5) and (-3, 2):

Transformations of Functions

Transformations shift, stretch, compress, or reflect the graph of a parent function.

• General form:

• h: Horizontal shift (opposite sign)

• k: Vertical shift

• a: Vertical stretch/compression; if negative, reflects across the x-axis

• Example: For :

  • Horizontal shift: Left 3 units

  • Vertical shift: Down 5 units

  • Vertical stretch by 2 and reflection across x-axis

  • Domain:

  • Range:

Modeling Functions

Expressing one quantity as a function of another using known formulas.

• Example: A string of length forms a circle. Express the area as a function of .

  • Circumference:

  • Area:

  • Function:

Quadratic Functions

Key Features of Parabolas

Quadratic functions have the form and their graphs are parabolas.

• Vertex: The turning point of the parabola.

  • x-coordinate:

  • y-coordinate: Substitute x into

• Axis of Symmetry: The vertical line (where h is the x-coordinate of the vertex).

• Intercepts:

  • y-intercept: Set

  • x-intercepts: Set and solve

• Opens Up or Down: If , opens up; if , opens down.

• Example: For :

  • ; opens up

  • Vertex: ;

  • Vertex: (3, -4)

  • Axis of symmetry:

  • y-intercept:

  • x-intercepts:

Systems of Equations

Solving Systems: Substitution and Elimination

Systems of equations can be solved using substitution or elimination methods.

• Substitution Method: Solve one equation for a variable, substitute into the other.

  • Example: ,

    • From second equation:

    • Substitute into first:

    • Then

    • Solution:

• Elimination Method: Add or subtract equations to eliminate a variable.

  • Example: ,

    • Multiply first equation by 2:

    • Compare with second:

    • Contradiction:

    • Conclusion: System is inconsistent (no solution)

Classification of Systems

Additional info: The above notes cover core precalculus topics relevant for a midterm, including equations, inequalities, functions, quadratics, and systems of equations. Practice problems are integrated as examples for each topic.