MODULE 7

• Quadratic inequalities in two variables

  • Always expressed in the general form: y < ax^2 + bx + c
  • y ≥ ax^2 + bx + c
  • y > ax^2 + bx + c
  • y ≤ ax^2 + bx + c
• Quadratic inequalities in two variables in standard form (vertex form)


  • f(x) > (x + 2)^2
  • h(x) ≥ (x - 3)^2 - 5
  • y ≤ (2x - 1)^2
  • 2x^2 - 7x + 3 ≥ y
• Illustrative example 1: Change 2x^2 + x + y ≥ -3x + 5 into its standard form

  1. Use Addition Property of Inequality

  2. Combine similar terms (left side terms) by using ideas of simplifying polynomials

  3. The standard form is y ≥ -2x^2 - 4x + 5
• Illustrative example 2: Express 2(r) + r^2 < -3r + 1 + f(r) into its standard form


  1. Use Addition Property of inequality

  2. Combine similar terms (left side terms) by using ideas of simplifying polynomials

  3. The standard form is r^2 + 5r - 1 < f(r)
• Illustrative example 3: Find the inequality as a function of time (t) when the ball is more than 64 m above the ground

  1. Substitute 64 to variable y

  2. Use Properties of Inequality

  3. Multiplying both sides by -1 will change the direction of inequality symbol into <

  4. The general inequality represented by the problem is 4t^2 - 24t + 64 < y or y > 4t^2 - 24t + 64
• Variable y


  To find the general inequality represented
• Problem

  4t^2 - 24t + 64 < y or y > 4t^2 - 24t + 64
• Quadratic inequalities in two variables can be expressed in standard form
• Given quadratic inequalities in two variables

  • x^2 > 2x + 2 + y
  • f(b) - b + b^2 < -1
  • 3p^2 + y ≥ -4
  • 8x^2 ≤ 72 + g(x)
  • y > 5x^2 - 25
  • y < 5(x + 1)^2 - 5
  • y ≤ x^2 - 2
  • y ≤ x^2 - 5(x - 2)
  • y > 5x^2 - 2x + 1
  • y ≥ 5x^2
• Change each quadratic inequality in two variables

  Using f(x), g(x), h(x), and P(x)
• Express the given quadratic inequalities into their standard form
• 3x^2 - y > 9x + 1
• f(r) - 2 + 5r^2 ≤ -3r + 1
• Express the following into standard form of quadratic inequalities

  1. t^2 + 3 > -8t + y

  2. (2x + 3)^2 + f(x) < x(x - 1)

  3. P(q) + q^2 - 3q ≥ 0

  4. -3p+p^2 ≤ 2(p + 5) + h(p)
• Mathematical sentences

  • a^2 - 6 ≤ y
  • x^2 + 3x ≥ -1
  • 4c^2 - 2c > 2c
• The square of twice x is less than -1

  Translated in standard form as (2x)^2 + 1 < 0
• Quadratic inequalities in two variables
• Quadratic inequality in two variables
• Determine the inequality that represents the height greater than 16 meters above the ground reached by the ball after t seconds

  • 16 + 6t - t^2 > P(x)
• Self-examples of quadratic inequalities in two variables expressed in standard form


  • y > x^2 + 5x + 6
  • 4a^2 + 3a ≥ f(a)
  • g(r) < 2r^2 - 5r + 6
  • h(x) ≤ x^2 - 9
• Express each quadratic inequality into quadratic equation in standard form

  y = x^2 + 5x
• Solve for the values of x in each quadratic equation

  1. x^2 - 5x - 50 = 0

  2. x^2 - 3x + 2 = 0

  3. 8x^2 = 32
• Determine the values of x in which the ball is above 72 m high


  How long would the ball be above that height
• Solving quadratic inequalities in two variables

  • Solutions can be illustrated visually through the use of the rectangular coordinate system
  • Solutions represent the shaded region of the graph
• Solve the quadratic inequality y ≥ x^2 - x - 6

  1. Transform the quadratic inequality to quadratic equation

  2. Construct table of values

  3. Plot the ordered pairs on the rectangular coordinate plane

  4. Determine the shaded region and the solution set
• The vertex of the parabola is (0.5, -6.32)
• All points on the shaded region of the graph and along the parabola are the solution set of the given quadratic inequality
• All points on the shaded region of the graph and along the parabola are the SOLUTION SET of the given quadratic inequality since the inequality symbol used is "≥".
• The given point (0, 1) is a solution to the inequality y > x^2 + 5x + 6.
• The given point (-2, 5) is not a solution to the inequality f(x) < 2x^2 - x + 3.
• The given point (-3, -4) is a solution to the inequality g(x) ≥ 9 - x^2.
• The given point (2, 0) is not a solution to the inequality 3x^2 < h(x) - 2x + 1.
• The given point (4, 4) is not a solution to the inequality 2(x - 1)^2 > P(x).
• The given point (0, 0) is a solution to the inequality y ≥ 3x^2.
• The ordered pair (-6, 7) does not satisfy the inequality x^2 + 10x ≥ y.
• The ordered pair (3, 6) satisfies the inequality x^2 + 10x ≥ y.
• The ordered pair (-6, -2) satisfies the inequality x^2 + 10x ≥ y.
• The ordered pair (-1, 6) satisfies the inequality x^2 + 10x ≥ y.
• The ordered pair (-5, 10) satisfies the inequality x^2 + 10x ≥ y.
• The ordered pair (-3, 4) satisfies the inequality x^2 + 10x ≥ y.