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Solving Quadratic Inequalities Representing Motion

Author: Sophia

what's covered
In this lesson, you will learn how to determine the solution set of a quadratic inequality in a given scenario. Specifically, this lesson will cover:

Table of Contents

1. Solving a Quadratic Inequality for an Object in Motion

Quadratic equations can be used to represent the path of objects as they go up in the air, and come back down due to the force of gravity.

EXAMPLE

Imagine a ball being thrown straight up into the air. For the first few seconds, the height of the ball will be increasing, but eventually, it will reach a maximum height and then begin to fall back down, eventually hitting the group. In relation to time, the graph might look something like this:



The graph of this curve can be represented by the following equation:

y equals short dash 16 x squared plus 80 x plus 6

did you know
The coefficients in this equation have important meanings within this context. short dash 16 x squaredcomes from the force of gravity on Earth, 80x represents the velocity of the ball, and 6 represents the height of the player hitting the ball.

We can use this equation to answer some questions about the height of the ball, and how long it is in the air.

EXAMPLE

Suppose we want to figure out when the ball is more than 70 feet in the air. We will use a quadratic inequality to represent this situation.

70 less than short dash 16 x squared plus 80 x plus 6

We use the < symbol to show that the height of the ball (represented by the quadratic equation) has a greater value than 70. We could have alternatively written our inequality with 70 on the right side, where the inequality symbol would be >, short dash 16 x squared plus 80 x plus 6 greater than 70.

To solve quadratic inequalities:
  1. Write as an equation set equal to zero.
  2. Solve the equation (using factoring, completing the square, the quadratic formula, etc.).
  3. Use solutions to create intervals on a number line.
  4. Identify intervals that make the inequality statement true.
Our first step is to write the inequality as an equation set equal to zero. To do this, we'll need to move the constant term to the other side:​

70 less than short dash 16 x squared plus 80 x plus 6 Write as an equation
70 equals short dash 16 x squared plus 80 x plus 6 Set equal to zero by subtracting 70 from both sides
0 equals short dash 16 x squared plus 80 x minus 64 Our equation

To solve this equation, we can use any method, although factoring and the quadratic formula are the most common. Factoring tends to involve fewer calculations, but it can take too long to identify such integers, so it is probably best to use the quadratic formula. We are going to solve this by trying both methods, factoring and using the quadratic formula. First, let's factor out a common factor between all coefficients.

0 equals short dash 16 x squared plus 80 x minus 64 Factor out -16
0 equals short dash 16 open parentheses x squared minus 5 x plus 4 close parentheses Factor the quadratic expression
0 equals short dash 16 open parentheses x minus 4 close parentheses open parentheses x minus 1 close parentheses Set each factor equal to zero
0 equals x minus 4 comma space space space space space space space space 0 equals x minus 1 Simplify each factor
x equals 1 comma space x equals 4 Our solutions

It was easy to factor this particular equation, but that's not always the case. If it looks complicated, you can always use the quadratic formula. Let's show how to find the same answers using this formula.

If we wanted to use the quadratic formula, we would get the same answers.

0 equals short dash 16 x squared plus 80 x minus 64 Identify coefficients a comma b, and c
a equals short dash 26 comma space b equals 80 comma space c equals short dash 64 Use quadratic formula
x equals fraction numerator short dash b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction Plug in a equals short dash 26 comma space b equals 80 comma space c equals short dash 64
x equals fraction numerator short dash 80 plus-or-minus square root of 80 squared minus 4 open parentheses short dash 16 close parentheses open parentheses short dash 64 close parentheses end root over denominator 2 open parentheses short dash 16 close parentheses end fraction Square 80 and multiply 4, -16, and -64
x equals fraction numerator short dash 80 plus-or-minus square root of 6400 minus 4096 end root over denominator 2 open parentheses short dash 16 close parentheses end fraction Simplify discriminant
x equals fraction numerator short dash 80 plus-or-minus square root of 2304 over denominator 2 open parentheses short dash 16 close parentheses end fraction Evaluate square root
x equals fraction numerator short dash 80 plus-or-minus 48 over denominator 2 open parentheses short dash 16 close parentheses end fraction Evaluate denominator
x equals fraction numerator short dash 80 plus-or-minus 48 over denominator short dash 32 end fraction Separate into two fractions, one with addition and one with subtraction
x equals fraction numerator short dash 80 minus 48 over denominator short dash 32 end fraction comma space space x equals fraction numerator short dash 80 plus 48 over denominator short dash 32 end fraction Evaluate numerators of both fractions
x equals fraction numerator short dash 128 over denominator short dash 32 end fraction comma space space x equals fraction numerator short dash 32 over denominator short dash 32 end fraction Simplify
x equals 4, and x equals 1 Our solutions

We use these two solutions to form intervals on the number line. Since we have two solutions, this breaks the number line into three sections:



Next, we choose a value that lies within each interval and plug that value into the inequality, to see if it yields a true statement. It is helpful to use the inequality with zero on the one side, so we can simply compare the values to zero. We can rewrite 70 less than short dash 16 x squared plus 80 x plus 6 as 0 less than short dash 16 x squared plus 80 x minus 64.

Using this rewritten inequality, choose test values within the three intervals. We need a test value that is less than 1, a test value that is between 1 and 4, and a test value greater than 5. Let's use 0, 2, and 5 and plug them into the inequality

Test Value bold 0 bold less than bold short dash bold 16 bold italic x to the power of bold 2 bold plus bold 80 bold italic x bold minus bold 64 Result Interval
0 table attributes columnalign left end attributes row cell 0 less than short dash 16 open parentheses 0 close parentheses squared plus 80 open parentheses 0 close parentheses minus 64 end cell row cell 0 less than 0 plus 0 minus 64 end cell row cell 0 less than short dash 64 end cell end table This inequality is FALSE, so this interval is NOT part of the solution. x less or equal than 1
2 table attributes columnalign left end attributes row cell 0 less than short dash 16 open parentheses 2 close parentheses squared plus 80 open parentheses 2 close parentheses minus 64 end cell row cell 0 less than short dash 16 open parentheses 4 close parentheses plus 160 minus 64 end cell row cell table attributes columnalign left end attributes row cell 0 less than short dash 64 plus 160 minus 64 end cell row cell 0 less than 32 end cell end table end cell end table This inequality is TRUE, so this interval is part of the solution. 0 less than x less than 4
5 table attributes columnalign left end attributes row cell 0 less than short dash 16 open parentheses 5 close parentheses squared plus 80 open parentheses 5 close parentheses minus 64 end cell row cell 0 less than short dash 16 open parentheses 25 close parentheses plus 400 minus 64 end cell row cell 0 less than short dash 400 plus 400 minus 64 end cell row cell 0 less than short dash 64 end cell end table This inequality is FALSE, so this interval is NOT part of the solution. x greater or equal than 4

This means that the ball is above 70 feet between 1 and 4 seconds after being launched into the air.

hint
You might have preferred to use the inequality written in factored form 0 less than short dash 16 open parentheses x minus 4 close parentheses open parentheses x minus 1 close parentheses, especially if the other side of the inequality is zero. This is because you can simply analyze the signs of each factor (positive or negative) to determine if the value of the quadratic will be positive or negative. From there, you can easily compare to zero without actually calculating exact values. Either way, you would get the same answers.


2. Quadratic Inequalities with No Solution

Let's take a look at an example where a quadratic inequality has no solutions.

EXAMPLE

We would like to know when the ball is above 120 feet. To answer this question, we follow the same steps as in our previous example, but with a different inequality to represent this new scenario.

120 less than short dash 16 x squared plus 80 x plus 6

The first step is writing this inequality as an equation set equal to zero.

120 less than short dash 16 x squared plus 80 x plus 6 Write as an equation
120 equals short dash 16 x squared plus 80 x plus 6 Set equal to zero by subtracting 120 from both sides
0 equals short dash 16 x squared plus 80 x minus 114 Our equation

Then we must solve 0 equals short dash 16 x squared plus 80 x minus 114. We will solve this using the quadratic formula.

x equals fraction numerator short dash b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction Plug in a equals short dash 26 comma space b equals 80 comma space c equals short dash 114
x equals fraction numerator short dash 80 plus-or-minus square root of 80 squared minus 4 open parentheses short dash 16 close parentheses open parentheses short dash 114 close parentheses end root over denominator 2 open parentheses short dash 16 close parentheses end fraction Square 80 and multiply 4, -16, and -114
x equals fraction numerator short dash 80 plus-or-minus square root of 6400 minus 7296 end root over denominator 2 open parentheses short dash 16 close parentheses end fraction Evaluate discriminant
x equals fraction numerator short dash 80 plus-or-minus square root of short dash 896 end root over denominator 2 open parentheses short dash 16 close parentheses end fraction Non-real solution

We cannot work any further, because we have a negative value underneath the square root. There is a special term for this value when working with the quadratic formula. We call this the discriminant. If the discriminant is negative, there is no real solution to the equation, because every real number square leads to a positive value.

Since there is no real solution, we know that the ball will never even reach 120 feet in the air.

summary
A quadratic equation can be used to model the path of an object rising and falling due to gravity. If x represents time and y represents height, we can solve a quadratic inequality for an object in motion, such as the time interval that an object is greater than or less than a certain height. There are also situations of quadratic inequalities with no solution. This is when the discriminant is negative.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License