Quadratic and Absolute Value Inequalities - Systems of Inequalities and Functions

Quadratic Inequalities in One Variable

  • Step 1: Rewrite the inequality so one side is equal to zero.
  • Step 2: Solve for x by factoring or using the quadratic formula.
  • Step 3: Test the intervals to find where the solutions are located.

Example:

x^2 - 3x \le 10

  1. Subtract 10 from both sides:
    x^2 - 3x - 10 \le 0
  2. Factor the quadratic expression:
    (x - 5)(x + 2) = 0
  3. Solve for x:
    x = 5, x = -2
  4. Test the intervals:
    • x < -2, test x = -3:
      (-3)^2 - 3(-3) = 9 + 9 = 18 \nleq 10, which is false.
    • -2 < x < 5, test x = 0:
      (0)^2 - 3(0) = 0 \le 10, which is true.
    • x > 5, test x = 6:
      (6)^2 - 3(6) = 36 - 18 = 18 \nleq 10, which is false.
  5. Solution:
    • The solutions are between -2 and 5, inclusive.
    • Interval notation: [-2, 5]

Quadratic Inequalities: Greater Than

Example:

x^2 + 3x > 4

  1. Subtract 4 from both sides:
    x^2 + 3x - 4 > 0
  2. Factor the quadratic expression:
    (x + 4)(x - 1) = 0
  3. Solve for x:
    x = -4, x = 1
  4. Test the intervals:
    • x < -4, test x = -5: (-5)^2 + 3(-5) = 25 - 15 = 10 > 4, which is true.
    • -4 < x < 1, test x = 0: (0)^2 + 3(0) = 0 > 4, which is false.
    • x > 1, test x = 2:
      (2)^2 + 3(2) = 4 + 6 = 10 > 4, which is true.
  5. Solution:
    • The solutions are to the left of -4 and to the right of 1.
    • Interval notation: (-\infty, -4) \cup (1, \infty)

General Observations

  • Less than or less than or equal to: solutions are generally in between.
  • Greater than or greater than or equal to: solutions are generally in separate sections.

Quadratic Inequalities in Two Variables

  • Must graph the function.
  • Determine key features: vertex, zeros, y-intercept.

Example:

y < x^2 - 2x - 3

  1. Find the vertex:
    • x = \frac{-b}{2a} = \frac{-(-2)}{2(1)} = 1
    • y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4
    • Vertex: (1, -4)
  2. Find the intercepts:
    • Set y = 0: x^2 - 2x - 3 = 0
      (x - 3)(x + 1) = 0
      x = 3, x = -1
    • Set x = 0: y = (0)^2 - 2(0) - 3 = -3
      • y-intercept: (0, -3)
      • x-intercept: (3, -1)
  3. Sketch the graph:
    • Use a dashed line because the inequality does not include "equal to."
  4. Test an ordered pair:
    • Test (0, 0): 0 < (0)^2 - 2(0) - 3 \implies 0 < -3, which is false.
  5. Solution:
    • Shade the outside of the parabola since (0,0) is not a solution.

Absolute Value Inequalities

  • Absolute value: distance of a number from zero.
    • |2| = 2
    • |-2| = 2

Solving Absolute Value Inequalities

  1. Isolate the absolute value expression.
  2. Rewrite as a compound inequality.
  3. Solve.

Less Than Inequality

|expression| < a \implies -a < expression < a

Greater Than Inequality

|expression| > a \implies expression < -a \text{ or } expression > a

Examples

Example 1 (Less Than)

2|10 + 4x| < 28

  1. Isolate the absolute value:
    |10 + 4x| < 14
  2. Rewrite as a compound inequality:
    -14 < 10 + 4x < 14
  3. Solve:
    • Subtract 10 from all sides: -24 < 4x < 4
    • Divide by 4: -6 < x < 1
  4. Solution:
    • Interval notation: (-6, 1)

Example 2 (Greater Than)

|4x - 7| + 8 \ge 17

  1. Isolate the absolute value:
    |4x - 7| \ge 9
  2. Rewrite as a compound inequality:
    4x - 7 \le -9 \quad \text{or} \quad 4x - 7 \ge 9
  3. Solve:
    • Add 7 to all sides: 4x \le -2 \quad \text{or} \quad 4x \ge 16
    • Divide by 4: x \le -\frac{1}{2} \quad \text{or} \quad x \ge 4
  4. Solution:
    • Interval notation: (-\infty, -\frac{1}{2}] \cup [4, \infty)

Systems of Inequalities

  • Graph each inequality separately.
  • Test an ordered pair to confirm the overlapping shaded region.

Systems with Constraints

  • Eliminate non-viable solutions.

Example

Local florist profit: 5x^2 + 10y \le 5000

Cost: 0.8x + y < 100

  • Viable solution area eliminates any negatives, because profit and cost cannot be negative.

Square Root Functions

y = -\sqrt{x - 1}

  • x-intercept: (1, 0)
  • No y-intercept.
  • Domain: x \in \mathbb{R} \text{ such that } x \ge 1 or [1, \infty)
  • Range: y \in \mathbb{R} \text{ such that } y \le 0 or (-\infty, 0]
  • Decreasing: The function decreases on the interval (1, \infty)

Cube Root Functions

y = -2 \sqrt[3]{x}

  • Domain: \mathbb{R} or (-\infty, \infty)
  • Range: \mathbb{R} or (-\infty, \infty)
  • Intercept: (0, 0)
  • End behavior:
    • As x \to -\infty, f(x) \to \infty
    • As x \to \infty, f(x) \to -\infty

Comparison of Functions

Polynomial Functions

  • Always continuous.
  • Always have a y-intercept, but not necessarily an x-intercept.
  • Equation: The terms have variables with positive whole number exponents and no variables in the denominator of any term.
  • Table: Look at the differences between the terms. If the differences are the same, it is that type of polynomial. (1st degree = Linear, 2nd Degree = Quadratic, 3rd Degree = Cubic, and so on).

Example

f(x) = x^3 - 8

  • To see how/if differences match, keep subtracting each adjacent term in the differences until they equal. First difference matches = Linear, Second difference matches = Quadratic, Third difference matches = Cubic.

Exponential Functions

  • Continuous.
  • Sharp curve and a horizontal asymptote.
  • Always have a y-intercept, but not necessarily an x-intercept.
  • Equation: The base is a positive number greater than zero but not equal to one, and the variable is an exponent.
  • Table: Constant change in x and a constant ratio in y.

Example

y = 4^x

Each time x increases: Y is being multiplied by 4

Rational Functions

  • Not continuous.
  • Vertical and horizontal asymptotes create gaps and branches on the graph.
  • The gaps and branches can sometimes be visible in the table form.
  • Equation: The ratio of two polynomial functions where the denominator cannot equal zero.

Example

f(x) = \frac{ 4x }{ (x^2 - 1)}

Logarithmic Functions

  • Inverse of exponential functions.
  • Continuous, but has a vertical asymptote and always have an x-intercept but not necessarily a y-intercept.
  • Equation: Will see the term "log" or "ln" for natural log.
  • Table: Look for a constant change in y and a constant ratio in x.
    • One dead giveaway for a log function is that a section of the graph is undefined. Consecutive values are undefined. This tells us that the function has a vertical asymptote at x zero because at one, two, three, and four now we all of a sudden have values that exist but not before the one.

Example

y = 3\ln{x}

Comparison of Two Functions

f(x) = -2^x

g(x) = -\sqrt{(x - 1)}

These are both upside down functions because they all have negative coefficients.

  • Both functions decrease as x increases.
  • They both decrease together on the interval from one to infinity: (1, \infty)
  • Both functions are negative.