Quadratic Inequalities - Examples, Exercises and Solutions

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

Suggested Topics to Practice in Advance

  1. Solution of a system of equations - one of them is linear and the other quadratic
  2. Quadratice Equations and Systems of Quadraric Equations
  3. Quadratic Equations System - Algebraic and Graphical Solution

Practice Quadratic Inequalities

Examples with solutions for Quadratic Inequalities

Exercise #1

Solve the following equation:

x^2+4>0

Video Solution

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

Exercise #2

Solve the following equation:

x^2+9>0

Video Solution

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

Exercise #3

Solve the following equation:

-x^2-9>0

Video Solution

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.

Exercise #4

Solve the following equation:

-x^2+2x>0

Video Solution

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

Exercise #5

Solve the following equation:

x^2-3x+4<0

Video Solution

Step-by-Step Solution

The problem requires us to solve the inequality x23x+4<0 x^2 - 3x + 4 < 0 .

To solve the inequality, we first consider the corresponding quadratic equation x23x+4=0 x^2 - 3x + 4 = 0 and find its roots.

Calculate the discriminant Δ \Delta :
Δ=b24ac=(3)2414=916=7\Delta = b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 4 = 9 - 16 = -7.

The discriminant Δ=7 \Delta = -7 is less than zero, indicating that the quadratic equation has no real roots. This implies that the quadratic expression x23x+4 x^2 - 3x + 4 does not change sign and is either always positive or always negative.

Next, evaluate the sign of x23x+4 x^2 - 3x + 4 . For x=0 x = 0 , the expression is 0230+4=4 0^2 - 3 \cdot 0 + 4 = 4 , which is positive. Therefore, the expression is always positive for all real x x .

Since x23x+4 x^2 - 3x + 4 is always positive, there is no x x for which x23x+4<0 x^2 - 3x + 4 < 0 holds true.

Therefore, the solution to the inequality is that there is no solution, which corresponds to option 4: "There is no solution."

Answer

There is no solution.

Exercise #6

Solve the following equation:

x^2-6x+8<0

Video Solution

Step-by-Step Solution

To solve this inequality x26x+8<0 x^2 - 6x + 8 < 0 , we first identify the roots of the equation x26x+8=0 x^2 - 6x + 8 = 0 .

Using the quadratic formula, where a=1 a = 1 , b=6 b = -6 , and c=8 c = 8 :

x=b±b24ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x=6±(6)241821 x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}

x=6±36322 x = \frac{6 \pm \sqrt{36 - 32}}{2}

x=6±42 x = \frac{6 \pm \sqrt{4}}{2}

x=6±22 x = \frac{6 \pm 2}{2}

The solutions are:

x=6+22=4andx=622=2 x = \frac{6 + 2}{2} = 4 \quad \text{and} \quad x = \frac{6 - 2}{2} = 2

The roots are x=2 x = 2 and x=4 x = 4 . These divide the number line into three intervals: (,2)(-\infty, 2), (2,4)(2, 4), and (4,)(4, \infty).

We test each interval to determine where the inequality is satisfied:

  • In the interval (,2)(-\infty, 2), select x=0 x = 0 . Then:
  • 026×0+8=8 0^2 - 6 \times 0 + 8 = 8 , which is greater than 0. Inequality not satisfied.

  • In the interval (2,4)(2, 4), select x=3 x = 3 . Then:
  • 326×3+8=918+8=1 3^2 - 6 \times 3 + 8 = 9 - 18 + 8 = -1 , which is less than 0. Inequality satisfied.

  • In the interval (4,)(4, \infty), select x=5 x = 5 . Then:
  • 526×5+8=2530+8=3 5^2 - 6 \times 5 + 8 = 25 - 30 + 8 = 3 , which is greater than 0. Inequality not satisfied.

Therefore, the solution to the inequality x26x+8<0 x^2 - 6x + 8 < 0 is the interval (2,4)(2, 4).

Thus, the correct answer is 2<x<4 2 < x < 4 .

Answer

2 < x < 4

Exercise #7

Solve the following equation:

-x^2+3x+4>0

Video Solution

Step-by-Step Solution

To solve this quadratic inequality, follow these steps:

  • Step 1: Solve the corresponding equation x2+3x+4=0-x^2 + 3x + 4 = 0 to find critical points.
  • Step 2: Test intervals between critical points to determine where the inequality holds.

Step 1: Solve the equation. The given quadratic is x2+3x+4=0-x^2 + 3x + 4 = 0. Let's rewrite it as x23x4=0x^2 - 3x - 4 = 0 by multiplying through by 1-1.

We use the quadratic formula where a=1a = 1, b=3b = -3, and c=4c = -4:

x=b±b24ac2a=(3)±(3)24(1)(4)2(1)x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-4)}}{2(1)}

x=3±9+162=3±252=3±52x = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm \sqrt{25}}{2} = \frac{3 \pm 5}{2}

The solutions to this equation are:

x1=3+52=4x_1 = \frac{3 + 5}{2} = 4 and x2=352=1x_2 = \frac{3 - 5}{2} = -1

Step 2: Determine where the expression is positive by checking intervals:

  • Interval (,1)(-\infty, -1): Choose x=2x = -2. Calculating the expression: (2)2+3(2)+4=46+4=6-(-2)^2 + 3(-2) + 4 = -4 - 6 + 4 = -6 (negative).
  • Interval (1,4)(-1, 4): Choose x=0x = 0. Calculating: 02+3(0)+4=4-0^2 + 3(0) + 4 = 4 (positive).
  • Interval (4,)(4, \infty): Choose x=5x = 5. Calculating: 52+3(5)+4=25+15+4=6-5^2 + 3(5) + 4 = -25 + 15 + 4 = -6 (negative).

The quadratic expression x2+3x+4-x^2 + 3x + 4 is positive in the interval (1,4)(-1, 4). Hence, for the inequality x2+3x+4>0-x^2 + 3x + 4 > 0, we have:

The solution to the inequality is 1<x<4 -1 < x < 4 .

Answer

-1 < x < 4

Exercise #8

Solve the following equation:

x^2-8x+12>0

Video Solution

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

Exercise #9

Solve the following equation:

x^2+4x>0

Video Solution

Step-by-Step Solution

To solve the inequality x2+4x>0 x^2 + 4x > 0 , we will:

  • Step A: Find the roots of the equation x2+4x=0 x^2 + 4x = 0 .
  • Step B: Factor the quadratic to x(x+4)=0 x(x + 4) = 0 , giving roots x=0 x = 0 and x=4 x = -4 .
  • Step C: Use these roots to break the number line into intervals: (,4) (-\infty, -4) , (4,0) (-4, 0) , and (0,) (0, \infty) .
  • Step D: Test an arbitrary value from each interval in the inequality x2+4x>0 x^2 + 4x > 0 .

Now, let's examine these intervals:

  • For (,4) (-\infty, -4) , choose x=5 x = -5 :
    (5)2+4(5)=2520=5>0 (-5)^2 + 4(-5) = 25 - 20 = 5 > 0 . This interval satisfies the inequality.
  • For (4,0) (-4, 0) , choose x=2 x = -2 :
    (2)2+4(2)=48=4<0 (-2)^2 + 4(-2) = 4 - 8 = -4 < 0 . This interval does not satisfy the inequality.
  • For (0,) (0, \infty) , choose x=1 x = 1 :
    12+4(1)=1+4=5>0 1^2 + 4(1) = 1 + 4 = 5 > 0 . This interval satisfies the inequality.

Therefore, the inequality x2+4x>0 x^2 + 4x > 0 holds true for the intervals (,4) (-\infty, -4) and (0,) (0, \infty) .

Therefore, the solution to the inequality is x<4,0<x x < -4, 0 < x .

Answer

x < -4,0 < x

Exercise #10

Solve the following equation:

x^2-9<0

Video Solution

Step-by-Step Solution

To solve the inequality x29<0 x^2 - 9 < 0 , we will perform the following steps:

  • Step 1: Factor the inequality x29=(x3)(x+3) x^2 - 9 = (x - 3)(x + 3) .
  • Step 2: Identify the critical values from the factored expression, which occur at x=3 x = 3 and x=3 x = -3 .
  • Step 3: Use these critical points to divide the number line into intervals: (,3) (-\infty, -3) , (3,3) (-3, 3) , and (3,) (3, \infty) .
  • Step 4: Test each interval to determine where the inequality holds:
    • For x=0 x = 0 in the interval (3,3) (-3, 3) , (03)(0+3)=9 (0 - 3)(0 + 3) = -9 , which satisfies <0 < 0 .
    • For x=4 x = -4 in (,3) (-\infty, -3) , (43)(4+3)=7 (-4 - 3)(-4 + 3) = 7 , which does not satisfy <0 < 0 .
    • For x=4 x = 4 in (3,) (3, \infty) , (43)(4+3)=7 (4 - 3)(4 + 3) = 7 , which does not satisfy <0 < 0 .

Therefore, the inequality x29<0 x^2 - 9 < 0 holds in the interval 3<x<3-3 < x < 3. This means any x x that falls between these values will satisfy the inequality.

The correct answer is 3<x<3\mathbf{-3 < x < 3}.

Answer

-3 < x < 3

Exercise #11

Solve the following equation:

x^2-16>0

Video Solution

Step-by-Step Solution

The objective is to find the values of x x such that the inequality x216>0 x^2 - 16 > 0 is satisfied.

Step 1: Factor the inequality expression.

The expression x216 x^2 - 16 can be factored using the difference of squares formula:

x216=(x4)(x+4) x^2 - 16 = (x - 4)(x + 4) .

Step 2: Determine the critical points.

Set the factors equal to zero to find the critical points:

  • x4=0 x - 4 = 0 gives x=4 x = 4 .
  • x+4=0 x + 4 = 0 gives x=4 x = -4 .

Step 3: Analyze the sign changes on the number line.

We test the intervals defined by the critical points 4-4 and 44 on a number line: (,4)(-∞, -4), (4,4)(-4, 4), (4,) (4, ∞) .

Choose a test point from each interval and substitute into the factored expression to check the sign.

  • For x=5 x = -5 (interval (,4)(-∞, -4)): (x4)(x+4)=(54)(5+4)=(9)(1)>0(x - 4)(x + 4) = (-5 - 4)(-5 + 4) = (-9)(-1) > 0.
  • For x=0 x = 0 (interval (4,4)(-4, 4)): (x4)(x+4)=(04)(0+4)=(4)(4)<0(x - 4)(x + 4) = (0 - 4)(0 + 4) = (-4)(4) < 0.
  • For x=5 x = 5 (interval (4,)(4, ∞)): (x4)(x+4)=(54)(5+4)=(1)(9)>0(x - 4)(x + 4) = (5 - 4)(5 + 4) = (1)(9) > 0.

Step 4: Extract the solution.

The inequality x216>0 x^2 - 16 > 0 holds true in the intervals where the product is positive, which are (,4)(4,) (-∞, -4) \cup (4, ∞) .

Therefore, the solution to the inequality is x<4 x < -4 or x>4 x > 4 .

The correct choice is x<4,4<x x < -4, 4 < x .

Answer

x < -4,4 < x

Exercise #12

Solve the following equation:

x^2-2x-8>0

Video Solution

Step-by-Step Solution

To solve the inequality x22x8>0 x^2 - 2x - 8 > 0 , we first need to find the roots of the related equation x22x8=0 x^2 - 2x - 8 = 0 .

Step 1: Factor the quadratic
The quadratic x22x8 x^2 - 2x - 8 can be factored as (x4)(x+2) (x - 4)(x + 2) because:

  • The product is 8 -8 and the sum is 2 -2 .
  • Expanding (x4)(x+2) (x - 4)(x + 2) , we get:
  • (x4)(x+2)=x2+2x4x8=x22x8(x - 4)(x + 2) = x^2 + 2x - 4x - 8 = x^2 - 2x - 8.

Step 2: Identify the roots
Set each factor to zero to find the roots:

  • x4=0x=4x - 4 = 0 \Rightarrow x = 4
  • x+2=0x=2x + 2 = 0 \Rightarrow x = -2

Step 3: Determine the intervals
The critical points divide the number line into three intervals: x<2x < -2, 2<x<4-2 < x < 4, and x>4x > 4.

Step 4: Test each interval
Choose test points from each interval to check where (x4)(x+2)>0 (x - 4)(x + 2) > 0 :

  • For x<2 x < -2 , take x=3 x = -3 :
    (34)(3+2)=(7)(1)=7>0(-3 - 4)(-3 + 2) = (-7)(-1) = 7 > 0
  • For 2<x<4-2 < x < 4 , take x=0 x = 0 :
    (04)(0+2)=(4)(2)=8<0(0 - 4)(0 + 2) = (-4)(2) = -8 < 0
  • For x>4 x > 4 , take x=5 x = 5 :
    (54)(5+2)=(1)(7)=7>0(5 - 4)(5 + 2) = (1)(7) = 7 > 0

Conclusion:
The solution to the inequality x22x8>0 x^2 - 2x - 8 > 0 is on the intervals x<2 x < -2 and x>4 x > 4 .

Final Answer:
The correct answer is: Answers (a) and (c)

Answer

Answers (a) and (c)

Exercise #13

Solve the following equation:

x^2-25<0

Video Solution

Step-by-Step Solution

To solve the inequality x225<0 x^2 - 25 < 0 , follow these steps:

  • Step 1: Factor the quadratic expression x225 x^2 - 25 . This is a difference of squares, which factors as (x5)(x+5)(x - 5)(x + 5).
  • Step 2: Identify the critical points where the expression equals zero, i.e., where (x5)(x+5)=0(x-5)(x+5) = 0. The solutions are x=5x = 5 and x=5x = -5.
  • Step 3: Determine the intervals defined by these critical points: (,5)(-∞, -5), (5,5)(-5, 5), and (5,)(5, ∞).
  • Step 4: Test each interval to see where the inequality (x5)(x+5)<0(x-5)(x+5) < 0 holds:
    • Choose a test point from (,5)(-∞, -5), such as x=6x = -6. Then, (65)(6+5)=(11)(1)>0(-6-5)(-6+5) = (-11)(-1) > 0. This interval does not satisfy the inequality.
    • Choose a test point from (5,5)(-5, 5), such as x=0x = 0. Then, (05)(0+5)=(5)(5)=25<0(0-5)(0+5) = (-5)(5) = -25 < 0. This interval satisfies the inequality.
    • Choose a test point from (5,)(5, ∞), such as x=6x = 6. Then, (65)(6+5)=(1)(11)>0(6-5)(6+5) = (1)(11) > 0. This interval does not satisfy the inequality.
  • Step 5: Conclude the solution. The inequality holds true in the interval (5,5)(-5, 5).

Therefore, the solution to the inequality x225<0 x^2 - 25 < 0 is 5<x<5-5 < x < 5.

Answer

-5 < x < 5

Exercise #14

Solve the following equation:

-x^2-25<0

Video Solution

Step-by-Step Solution

To solve the inequality x225<0-x^2 - 25 < 0, we start by simplifying it. Rearrange the inequality:

x2<25-x^2 < 25

Multiplying through by 1-1 (reversing the inequality sign), we have:

x2>25x^2 > -25

Since x2x^2 is always non-negative for all real numbers (i.e., x20x^2 \geq 0), the smallest value x2x^2 can take is 00. Therefore, x2x^2 is always greater than 25-25, since any non-negative number is greater than a negative number. Thus, this inequality holds true for all real values of xx.

Therefore, the solution to the inequality x225<0-x^2 - 25 < 0 is

All values of xx.

Answer

All values of x x

Exercise #15

Solve the following equation:

x^2+6x>0

Video Solution

Step-by-Step Solution

To solve the inequality x2+6x>0 x^2 + 6x > 0 , follow these steps:

  • Step 1: Write the inequality in factored form.
    Express x2+6x x^2 + 6x as x(x+6) x(x + 6) .
  • Step 2: Identify the roots of the equation x(x+6)=0 x(x + 6) = 0 .
    The roots are x=0 x = 0 and x=6 x = -6 .
  • Step 3: Determine the sign of x(x+6) x(x + 6) in each interval divided by the roots.
  • Step 4: Test three intervals: x<6 x < -6 , 6<x<0 -6 < x < 0 , and x>0 x > 0 .

For x<6 x < -6 :
Pick a value such as x=7 x = -7 . Substituting, x(x+6)=(7)(7+6)=(7)(1)=7>0 x(x + 6) = (-7)(-7 + 6) = (-7)(-1) = 7 > 0 .
Thus, x2+6x>0 x^2 + 6x > 0 for x<6 x < -6 .

For 6<x<0-6 < x < 0 :
Pick a value such as x=3 x = -3 . Substituting, x(x+6)=(3)(3+6)=(3)(3)=9<0 x(x + 6) = (-3)(-3 + 6) = (-3)(3) = -9 < 0 .
Thus, x2+6x<0 x^2 + 6x < 0 for 6<x<0-6 < x < 0 .

For x>0 x > 0 :
Pick a value such as x=1 x = 1 . Substituting, x(x+6)=(1)(1+6)=1×7=7>0 x(x + 6) = (1)(1 + 6) = 1 \times 7 = 7 > 0 .
Thus, x2+6x>0 x^2 + 6x > 0 for x>0 x > 0 .

Therefore, the solution to the inequality is x<6 x < -6 or x>0 x > 0 .

Thus, the correct answer is x<6,0<x x < -6, 0 < x .

Answer

x < -6,0 < x