Quadratic Inequalities | Mathematics for ACT PDF Download

Introduction

• Quadratic inequalities can be derived from quadratic equations. The word “quadratic” comes from the word “quadrature”, which means "square" in Latin. From this, we can define quadratic inequalities as second-degree inequation. Here, we first define a quadratic equation. The general form of a quadratic equation is ax² + bx + c = 0. Further if the quadratic polynomial ax² + bx + c is not equal to zero, then they are either ax² + bx + c > 0, or ax² + bx + c < 0, and are referred as quadratic inequalities.

What Do You Mean By Quadratic Inequalities?

• The quadratic inequality is a second-degree expression in x and has a greater than (>) or lesser than (<) inequality. the quadratic inequality has been derived from the quadratic equation ax² + bx + c = 0. Let us check the definition of quadratic inequality, the standard form, and the examples of quadratic inequalities.

Definition

• If a quadratic polynomial in one variable is less than or greater than some number or any other polynomial (with a degree less than or equal to 2), then it is said to be a quadratic inequality. The difference between a quadratic equation and a quadratic inequality is that the quadratic equation is equal to some number while quadratic inequality is either less than or greater than some number. Some examples of quadratic inequalities in one variable are:
  • x² + x - 1 > 0
  • 2x² - 5x - 2 > 0
  • x² + 2x - 1 < 0

Standard Form

• The standard form of quadratic inequalities in one variable is almost the same as the standard form of a quadratic equation. The only difference is that the quadratic equation has an "equal to" sign in it while a quadratic inequality has a "greater than" or "less than" sign (> or <). The standard form of quadratic inequality can be represented as:
  ax² + bx + c > 0
  or
  ax² + bx + c > 0

Example of Quadratic Inequality

• Now, consider the scenario where you want to build a rectangular house with a length equal to two units more than twice its breadth. If you don't want the floor area of the house to be more than 1500 ft², what length and breadth can you consider?
• You know that the area of a rectangle is length times its breadth. Hence, the area of the house is (2 + 2x)x = 2x² + 2x, where x is the breadth of the rectangular house. Now, we know that the area cannot exceed 1500 ft². Thus, the quadratic inequality for the above scenario is as follows.
  2x² + 2x < 1500

How To Solve Quadratic Inequalities?

• Solving a quadratic inequality means to find the values of x which satisfy the given condition of the question. A quadratic second degree equation ax² + bx + c = 0 can have maximum 2 values of x. But a quadratic inequality can have more than 2 values. It can have infinite values of x which satisfy the condition ax² + bx + c < 0 or ax² + bx + c > 0. Solving a quadratic inequation means finding the range of values of x.
• Now consider a quadratic expression ax² + bx + c. We can write the quadratic expression in the form of (x - α)(x - β) and α < β. Using the number line method the solution for the quadratic inequality can be expressed as follows.
  

Quadratic Inequalities on Line Graph

• It means that if ax² + bx + c > 0, then x can take values between - ∞ to α and β to +∞.
• If x² + bx + c > 0, then x ∈ (-∞ , α) U (β, + ∞)
• If x² + bx + c < 0, then x can take values between α and β.
• If x² + bx + c < 0, then x ∈ (α, β)

Let's take a quadratic inequality x² - 1 > 0. Here the expression x² - 1 > 0 can be factorized as (x - 1)(x + 1) > 0. This gives the values of α = -1 and β = 1. Hence, we obtain the range of x as x ∈ (-∞ , -1) U (1, + ∞)

If the quadratic inequality is x² - 1 < 0. The expression x2 - 1 < 0, can be factorized as (x - 1)(x + 1) < 0. This gives α = -1 and β = 1. Therefore, the range of x is x ∈ (-1, 1)

If the quadratic inequality is x² - 1 > 0 (where it shows the quadratic inequality is greater than or equal to zero). The expression x² - 1 > 0 can be factorized as (x - 1)(x + 1) > 0. Here we obtain α = -1 and β = 1, and the range of x is x ∈ {-∞ , -1] U [+1, + ∞}

If the quadratic inequality is x² - 1 < 0 (where it shows the quadratic inequality is less than or equal to zero). The expression x2 - 1 < 0 is factorized as (x - 1)(x + 1) < 0. Here the roots of the expression are α = -1 and β = 1, and the range of x is x ∈ [-1, +1]

Notations Used In Quadratic Inequalities

• The notation of greater than (>) and lesser. than (< ) is often used in quadratic equations. The quadratic equation ax² + bx + c = 0 is written as a quadratic equation by replacing equals to the symbol (=) with greater than or lesser than inequality. The general format of a quadratic inequality is ax² + bx + c > 0, or ax² + bx + c < 0. Further, let us check some of the other important notations used in quadratic inequalities.
  • ( ) → Open Brackets
  • [ ] → Closed Brackets
  • o → Open Value( x cannot take this value)
  • • → Closed Value( x can take this value)
  • (-1, 1) → x cannot take value -1 and 1.
  • [-1, 1) → x can take value -1 but not 1.
  • (-1, 1] → x cannot take value -1 but it can take value 1.
  • [-1, 1] → x can take both -1 and 1 values