Solving Equations and Inequalities
Contents
Inequalities
As opposed to an equation, an inequality is an expression that states that two quantities are unequal or not equivalent to one another. In most cases we use inequalities in real life more than equations (i.e. this shirt costs $2 more than that one).
Given that a and b are real numbers, there are four basic inequalities:
- a < b
- a is "less than" b:
- Example: 2 < 4 ; -3 < 0; etc.
- a > b
- a is "greater than" b:
- Example: -2 > -4 ; 3 > 0 ; etc.
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \le b}
- a is "less than or equal to" b:
- Example: If we know that , then we can conclude that x is equal to any value less than 7, including 7 itself.
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ge b}
- a is "greater than or equal to" b:
- Example: Conversely, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \ge 7} , then x is equal to any value greater than 7, including 7 itself.
Properties of Inequalities
Just as there are four properties of equality, there are also four properties of inequality:
Addition Property of Inequality
If a, b, and c are real numbers such that a > b, then a + c > b + c. Conversely, if a < b, then a + c < b + c.
Subtraction Property of Inequality
If a, b, and c are real numbers such that a > b, then a - c > b - c. Conversely, if a < b, then a - c < b - c.
Multiplication Property of Inequality
If a, b, and c are real numbers such that a > b and c > 0, then ac (or a * c) > bc (or b * c). Conversely, if a < b and c > 0, then ac < bc. (Note that if c = 0, then both sides of the inequality are in fact equal.) We will also review cases where c is less than zero later in the lesson.
Division Property of Inequality
If a, b, and c are real numbers such that a > b, and c > 0, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {a}{c} > \frac {b}{c}} . Conversely under the same conditions, if a < b, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {a} {c} < \frac {b}{c}} . As with the Multiplication Property, there are special cases that will be discussed later when c < 0.
Note that all four properties also work with inequalities where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \le b}
or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ge b}
.
Trichotomy Property
The statements above form the basis of Trichotomy Property:
Given any two real numbers a and b, then only one of the following statements must hold true:
- a < b
- a = b
- a > b
So, if we are given any two unknown real-number values, then any one of the three statements will hold true.
Solving Inequalities
Solving algebraic inequalities is more or less identical to solving algebraic equations. Consider the following example:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x + 4 \le 3x - 7}
Although it may be an inequality, we can use the Properties of Inequality stated above to solve. Start by subtracting 2x from both sides:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x - 2x + 4 \le 3x - 2x - 7}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 \le x - 7}
Finish by adding 7 to both sides.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 + 7 \le x - 7 + 7}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 11 \le x}
This can be rewritten as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \ge 11} . To check, substitute any value greater than or equal to 11. However, in order to satisfy the Trichotomy Property, we'll substitute three different values: 10, 11, and 12.
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x + 4 \le 3x - 7} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x + 4 \le 3x - 7} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x + 4 \le 3x - 7} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2(10) + 4 \le 3(10) - 7} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2(11) + 4 \le 3(11) - 7} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2(12) + 4 \le 3(12) - 7} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 24 \le 23} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 26 \le 26} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 28 \le 29} |
Ten is incorrect, whereas eleven and twelve satisfy the solution. Therefore, the solution set - the set of all answers which satisfy the original inequality - is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \ge 11} . Written in set notation, the answer is {Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x | x \ge 11} }. This is read as "the set of all x such that x is greater than or equal to 11".
Special Cases - A variable in the denominator
For example, consider the inequality
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2}{x-1}<2\,}
In this case one cannot multiply the right hand side by (x-1) because the value of x is unknown. Since x may be either positive or negative, you can't know whether to leave the inequality sign as <, or reverse it to >. The method for solving this kind of inequality involves four steps:
- Find out when the denominator is equal to 0. In this case it's when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=1} .
- Pretend the inequality sign is an = sign and solve it as such: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2}{x-1}=2\,} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2} .
- Plot the points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2} on a number line with an unfilled circle because the original equation included < (it would have been a filled circle if the original equation included <= or >=). You now have three regions: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x<1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1<x<2} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x>2} .
- Test each region independently. in this case test if the inequality is true for 1<x<2 by picking a point in this region (e.g. x=1.5) and trying it in the original inequation. For x=1.5 the original inequation doesn't hold. So then try for 1>x>2 (e.g. x=3). In this case the original inequation holds, and so the solution for the original inequation is 1>x>2.
Practice Problems
Special Cases
Let us suppose that we were given this inequality to solve:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3x + 8 \ge 26}
Using the same steps as above, start by subtracting 8 from both sides.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3x + 8 - 8 \ge 26 - 8}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3x \ge 18}
Now divide both sides by -3.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3x/-3 \ge 18/-3}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \ge -6}
Check this solution by substituting three numbers. We'll use -7, -6, and -5.
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3x + 8 \ge 26} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3x + 8 \ge 26} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3x + 8 \ge 26} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3(-7) + 8 \ge 26} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3(-6) + 8 \ge 26} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3(-5) + 8 \ge 26} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 29 \ge 26} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 26 \ge 26} |
Wait, what happened? -7 and -6 satisfy the inequality, yet -7 is non-inclusive in the solution set! And -5, which is greater than -6, does not satisfy the inequality!
This is a special case involved in solving inequalities. Because the coefficient of the x-term was negative, the constant on the other side (26, which became 18 and then -6) switched signs. In order to attain a valid solution if a negative number is divided, we need to switch the sign in order to make sure that the solution set is correct. Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \ge -6} becomes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \le -6} , or more specifically, {Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x | x \le -6} }.
Practice Problems
Graphing Solutions
Because inequalities have multiple solutions, we need to be able to represent them graphically. In order to do so, we use the number line, depicted below:
<--|--|--|--|--|--|--|--|--|--|--|-->
-5 -4 -3 -2 -1 0 1 2 3 4 5
There are two ways of graphing solutions; however, each one is unique depending on the nature of the inequality. If, for example, the inequality contains the < or > sign, we use an open circle ("O") and place it on (or above) the corresponding position on the number line, then draw another line either left or right of the solution (based on the sign) to indicate the infinite number of solutions in the set. For example, if we were to graph x < 4:
<-----------------------------O
<--|--|--|--|--|--|--|--|--|--|--|-->
-5 -4 -3 -2 -1 0 1 2 3 4 5
The open circle indicates that 4 is not included in the solution set; however, all values less than 4 satisfy the solution.
If an inequality contains a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \le} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ge} , then a closed circle (it will be depicted here with a *) is placed on or above the corresponding position on the number line. Hence, the solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \ge -2} is graphed as follows.
*----------------------->
<--|--|--|--|--|--|--|--|--|--|--|-->
-5 -4 -3 -2 -1 0 1 2 3 4 5
The asterisk (*) indicates that -2 is inclusive in the solution set, as are all values greater than -2.
Practice Problems
Graph the following inequalities on a number line:
1. x > 4
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \le 1}
3. -1 > x
Solutions
1.
O----->
<--|--|--|--|--|--|--|--|--|--|--|-->
-5 -4 -3 -2 -1 0 1 2 3 4 5
2.
<--------------------*
<--|--|--|--|--|--|--|--|--|--|--|-->
-5 -4 -3 -2 -1 0 1 2 3 4 5
(Note that your actual graph should have a closed circle in place of the dot. If in doubt, simply draw a circle and color it black.)
3. Be careful here; you need to rearrange the inequality first before graphing. When rewritten, the inequality becomes x < -1:
<--------------O
<--|--|--|--|--|--|--|--|--|--|--|-->
-5 -4 -3 -2 -1 0 1 2 3 4 5
Lesson Review
An inequality is a statement justifying that two quantities are not equal to each other. There are four cases of inequalities, two of which allow for equality (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \le b} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ge b} ). The four properties of inequality, which are more or less parallel to the properties of equality, can be used to solve simple inequalities. The only exceptions are in multiplication and division, where the signs must be reversed if both sides are multiplied or divided by a negative number. Finally, the solution set of an inequality can be graphed on the number line with either an open circle ("O") or a closed circle ("*"), depending on the original sign used.
Lesson Quiz
1. What is an inequality? List the four possible cases of inequalities.
2. State the Trichotomy Property in your own words.
3. Solve the following and graph all solutions:
a. 4x + 3 > 7 b. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x - 4 \le -x + 1}
c. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {-12c}{3} \ge -24}
d. -x + 4 > 3x
Quiz Answers
1. An inequality states that two quantities are not equal. The four cases of inequalities: a < b, a > b, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \le b} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ge b} .
2. Answers will vary. Just make sure you didn't copy and paste the actual definition.
3a. x > 1
O-------------->
<--|--|--|--|--|--|--|--|--|--|--|-->
-5 -4 -3 -2 -1 0 1 2 3 4 5
3b. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \le \frac {5}{3}}
<----------------------*
<--|--|--|--|--|--|--|--|--|--|--|-->
-5 -4 -3 -2 -1 0 1 2 3 4 5
3c. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \le 6}
<--------------------------------------*
<--|--|--|--|--|--|--|--|--|--|--|--|--|-->
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
3d. x < 1
<--------------------O
<--|--|--|--|--|--|--|--|--|--|--|-->
-5 -4 -3 -2 -1 0 1 2 3 4 5
Resources and Examples
General Resources
- Solving Equations and Inequalities, Paul's Online Notes
- Solving Equations and Inequalities, Khan Academy
- Rules for Solving Inequalities, Math Is Fun
Solving Different Types of Equations
- College Algebra from Lumen Learning:
- Solving Absolute Value Equations, Cliff's Notes
Solving Different Types of Inequalities
- Solving Linear Inequalities, patrickJMT on YouTube
- Solving Quadratic Inequalities, patrickJMT on YouTube
- Solving Quadratic Inequalities: More Examples, patrickJMT on YouTube
- Solving Absolute Value Inequalities, patrickJMT on YouTube
- Solving Inequalities with Radicals, University of Michigan Math Prep
NOTE: The same methods for solving equations can be used for solving inequalities. However, multiplying or dividing by a negative number on both sides of an inequality reverses the inequality sign. Examples of this:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -x < 7 \implies (-1)(-x) > (-1)(7) \implies x > -7 }
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6 - y \ge x \implies y - 6 \le -x \implies y \le 6 - x}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2x > 24 \implies (-2x)/(-2) < 24/(-2) \implies x < -12 }
