Quadratic Inequalities - Examples, Exercises and Solutions

Understanding Quadratic Inequalities

Complete explanation with examples

The quadratic inequality shows us in which interval the function is positive and in which it is negative - according to the inequality symbol. To solve quadratic inequalities correctly, it is convenient to remember two things:

  1. Set of positivity and negativity of the function:
    Set of positivity - represents the XXs in which the graph of the parabola is above the XX axis, with YY value positive.
    Set of negativity - represents the XXs in which the graph of the parabola is below the XX axis, with YY value negative.
  2. Dividing by a negative term - reverses the sign of the inequality.

Method to solve the quadratic inequality:

  1. We will carry out the transposition of members and isolate the quadratic equation until one side equals 0. Remember that when we divide by a negative term, the inequality is reversed.
  2. Let's draw a diagram of the parabola - placing points of intersection with the XX axis and identifying the maximum and minimum of the parabola.
  3. Let's calculate the corresponding interval according to the exercise and the diagram.
    Quadratic equation >0>0∶ Set of positivity
    Quadratic equation <0<0∶ Set of negativity
Detailed explanation

Practice Quadratic Inequalities

Test your knowledge with 7 quizzes

Solve the following equation:

\( x^2-2x-8>0 \)

Examples with solutions for Quadratic Inequalities

Step-by-step solutions included
Exercise #1

Solve the following equation:

x^2+4>0

Step-by-Step Solution

To solve this problem, let's examine the inequality x2+4>0 x^2 + 4 > 0 .

The expression x2+4 x^2 + 4 consists of two terms: x2 x^2 and 4 4 . Notice that:

  • The term x2 x^2 is always non-negative, which means x20 x^2 \geq 0 for any real number x x .
  • The constant term 4 4 is positive.

Combining these observations, we see that:

  • Since x2 x^2 is non-negative, x2+44 x^2 + 4 \geq 4 .
  • Therefore, x2+4 x^2 + 4 is always greater than zero, as adding 4 to a non-negative number will always yield a positive result.

Thus, there are no values of x x for which the expression x2+4 x^2 + 4 is zero or negative. Instead, the expression is always positive for all real numbers x x .

Therefore, the solution to the inequality x2+4>0 x^2 + 4 > 0 is all values of x x .

Answer:

All values of x x

Video Solution
Exercise #2

Solve the following equation:

x^2+9>0

Step-by-Step Solution

Let's explore this problem step-by-step:

The inequality given is x2+9>0 x^2 + 9 > 0 .

1. To understand this inequality, we start by considering the expression x2 x^2 . We know that for any real number x x , x20 x^2 \geq 0 . This means x2 x^2 is always non-negative.

2. Since x20 x^2 \geq 0 for every real number, adding 9 to x2 x^2 will necessarily make the expression greater than zero, because a non-negative number plus a positive number gives a positive result: x2+99>0 x^2 + 9 \geq 9 > 0 .

3. Therefore, the inequality x2+9>0 x^2 + 9 > 0 holds true for all real numbers x x . There is no value of x x that makes the left side equal to or less than zero.

4. Thus, the solution to the inequality is that it holds for all values of x x .

Consequently, the correct choice from the options provided is:

  • All values of x x

Therefore, the solution is that the inequality x2+9>0 x^2 + 9 > 0 is true for all values of x x .

Answer:

All values of x x

Video Solution
Exercise #3

Solve the following equation:

-x^2+2x>0

Step-by-Step Solution

To solve the inequality x2+2x>0-x^2 + 2x > 0, we begin by considering the corresponding equation x2+2x=0-x^2 + 2x = 0.

First, factor the quadratic equation:

  • Rearrange the terms: x2+2x=0-x^2 + 2x = 0 becomes x(2x)=0x(2 - x) = 0.
  • This gives us the roots x=0x = 0 and x=2x = 2.

These roots divide the number line into three intervals: x<0x < 0, 0<x<20 < x < 2, and x>2x > 2.

We need to test these intervals to determine where the inequality holds:

  • For x<0x < 0, choose a test point like x=1x = -1: the expression (1)2+2(1)=12=3-(-1)^2 + 2(-1) = -1 - 2 = -3, which is not greater than zero.
  • For 0<x<20 < x < 2, choose a test point like x=1x = 1: the expression (1)2+2(1)=1+2=1-(1)^2 + 2(1) = -1 + 2 = 1, which is greater than zero.
  • For x>2x > 2, choose a test point like x=3x = 3: the expression (3)2+2(3)=9+6=3-(3)^2 + 2(3) = -9 + 6 = -3, which is not greater than zero.

Thus, the inequality x2+2x>0-x^2 + 2x > 0 is satisfied for the interval 0<x<20 < x < 2.

Therefore, the solution to the inequality is 0<x<2\mathbf{0 < x < 2}, which corresponds to choice 2 in the given options.

Answer:

0 < x < 2

Video Solution
Exercise #4

Solve the following equation:

-x^2-9>0

Step-by-Step Solution

To solve this quadratic inequality, x29>0 -x^2 - 9 > 0 , we will follow these steps:

  • Step 1: Identify the quadratic expression x29 -x^2 - 9 .
  • Step 2: Attempt transformation and determine when the expression x29 -x^2 - 9 , can be greater than zero.

Let's analyze the equation:

Rewrite the inequality:
x29>0-x^2 - 9 > 0

Add 9 to both sides:
x2>9-x^2 > 9

Multiply the entire inequality by 1-1 and remember to reverse the inequality sign:
x2<9x^2 < -9

Observe the inequality x2<9x^2 < -9:
Note that x2x^2, being a square of any real number, is always greater than or equal to zero.

As x2x^2 cannot be less than negative nine for any real number xx, the inequality has no solution in the realm of real numbers.

Therefore, the correct answer is:

There is no solution.

Answer:

There is no solution.

Video Solution
Exercise #5

Solve the following equation:

x^2-8x+12>0

Step-by-Step Solution

Let's proceed to solve the inequality x28x+12>0 x^2 - 8x + 12 > 0 .

  • Start by factoring the quadratic: x28x+12 x^2 - 8x + 12 .
  • Identify the factors of 12 that sum to 8: 6 6 and 2 2 . This results in: (x6)(x2)=0 (x - 6)(x - 2) = 0 .

The factorization gives us the critical points x=6 x = 6 and x=2 x = 2 . These points divide the number line into three intervals: x<2 x < 2 , 2<x<6 2 < x < 6 , and x>6 x > 6 .

Now, we evaluate the sign of the product (x6)(x2) (x - 6)(x - 2) in each interval:

  • For x<2 x < 2 : Both (x6) (x - 6) and (x2) (x - 2) are negative, so their product is positive.
  • For 2<x<6 2 < x < 6 : (x2) (x - 2) is positive, (x6) (x - 6) is negative, so their product is negative.
  • For x>6 x > 6 : Both (x6) (x - 6) and (x2) (x - 2) are positive, so their product is positive.

The inequality (x6)(x2)>0 (x - 6)(x - 2) > 0 holds for x<2 x < 2 and x>6 x > 6 .

Thus, the solution to the inequality x28x+12>0 x^2 - 8x + 12 > 0 is x<2 x < 2 or x>6 x > 6 .

Therefore, the correct answer is x<2,6<x \boxed{x < 2, 6 < x} .

Answer:

x < 2,6 < x

Video Solution

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