B6, B7

Equations & Inequalities

Solving Absolute Value Equations and Inequalities

Absolute value equations and inequalities are a key topic in College Algebra, involving expressions that measure the distance from zero on the number line. Solving these requires understanding both algebraic manipulation and graphical interpretation.

• Absolute Value Equation: An equation of the form $|x| = a$ has two solutions: $x = a$ and $x = -a$.

• Absolute Value Inequality: For $|x| < a$, the solution is $-a < x < a$. For $|x| > a$, the solution is $x < -a$ or $x > a$.

• Key Steps:

  1. Isolate the absolute value expression.

  2. Set up two cases: one for the positive and one for the negative scenario.

  3. Solve each case separately.

  4. Check for extraneous solutions, especially when variables are inside the absolute value.

• Important Note: When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality sign.

Example: Solve $|2x - 3| < 5$.

• Set up: $-5 < 2x - 3 < 5$

• Add 3: $-2 < 2x < 8$

• Divide by 2: $-1 < x < 4$

Functions

Absolute Value Functions and Their Graphs

The absolute value function $f(x) = |x|$ is a basic example of a piecewise function, and its graph is a "V" shape centered at the origin. More complex absolute value functions can be graphed and analyzed using tables of values and piecewise definitions.

• Definition: The absolute value of a number $a$, denoted $|a|$, is its distance from zero on the number line.

• Piecewise Representation:

  • $|x| = egin{cases} x & x \\geq 0 \\ -x & x < 0 \end{cases}$

• Graphing: Create a table of values for $x$ and $f(x)$, plot the points, and connect them to form the graph.

• Example: For $f(x) = |x|$, the graph is a "V" with vertex at $(0,0)$.

Piecewise Functions from Absolute Value

Absolute value functions can be rewritten as piecewise functions, which are defined by different expressions over different intervals.

• Example: $f(x) = |x - 3|$ can be written as:

  • $f(x) = x - 3$ for $x \\geq 3$

  • $f(x) = -(x - 3)$ for $x < 3$

• Graph: The graph has a vertex at $(3,0)$ and is symmetric about $x = 3$.

Graphs of Equations

Graphing and Transforming Absolute Value Functions

Transformations of absolute value functions involve shifting, stretching, compressing, or reflecting the graph. These changes are represented algebraically and visually.

• Vertical Shifts: $f(x) = |x| + k$ shifts the graph up by $k$ units.

• Horizontal Shifts: $f(x) = |x - h|$ shifts the graph right by $h$ units.

• Reflections: $f(x) = -|x|$ reflects the graph over the $x$-axis.

• Stretch/Compression: $f(x) = a|x|$ stretches the graph vertically if $|a| > 1$ and compresses it if $0 < |a| < 1$.

• Vertex: The vertex of $f(x) = a|x - h| + k$ is at $(h, k)$.

Example: $f(x) = -|x - 3| - 2$ is reflected, shifted right by 3, and down by 2.

Equations & Inequalities

Solving Absolute Value Equations and Inequalities Graphically

Graphs can be used to find solutions to absolute value equations and inequalities by identifying points of intersection and intervals where the function meets the required conditions.

• Intersection Points: Solutions to $|x| = a$ are where the graph of $y = |x|$ meets $y = a$.

• Intervals: Solutions to $|x| < a$ are the $x$-values where the graph of $y = |x|$ is below $y = a$.

Functions

Identifying One-to-One Functions

A function is one-to-one if each output is produced by exactly one input. This property is important for determining invertibility.

• Horizontal Line Test: If any horizontal line crosses the graph more than once, the function is not one-to-one.

• Example: $f(x) = |x|$ is not one-to-one because $f(-a) = f(a)$ for $a \\neq 0$.

Tables

Comparison of Absolute Value Equations and Inequalities

The following table summarizes the solution methods for absolute value equations and inequalities:

Type	General Form	Solution
Equation	$|x| = a$	$x = a$ or $x = -a$
Inequality (less than)	$|x| < a$	$-a < x < a$
Inequality (greater than)	$|x| > a$	$x < -a$ or $x > a$

Additional Practice

Practice Problems and Applications

Practice problems reinforce the concepts of solving, graphing, and transforming absolute value equations and inequalities. These include creating tables of values, graphing piecewise functions, and solving equations both algebraically and graphically.

• Example: Solve $|x + 2| = 5$.

  • Case 1: $x + 2 = 5 \Rightarrow x = 3$

  • Case 2: $x + 2 = -5 \Rightarrow x = -7$

• Example: Write $f(x) = |x - 2| - 3$ as a piecewise function:

  • $f(x) = x - 2 - 3 = x - 5$ for $x \geq 2$

  • $f(x) = -(x - 2) - 3 = -x + 2 - 3 = -x - 1$ for $x < 2$

Additional info: These notes cover objectives B6 and B7 from a College Algebra course, focusing on absolute value equations, inequalities, their graphical and symbolic representations, and piecewise functions. The content is suitable for exam preparation and reinforces foundational algebraic skills.