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Study Guides > Intermediate Algebra

3A.1 Single- and Multi-Step Inequalities

3A.1 Learning Objectives

  • Represent inequalities on a number line
  • Represent inequalities using interval notation
  • Use the addition and multiplication properties to solve algebraic inequalities
  • Express solutions to inequalities graphically, with interval notation, and as an inequality
Sometimes there is a range of possible values to describe a situation. When you see a sign that says “Speed Limit 25,” you know that it doesn’t mean that you have to drive exactly at a speed of 25 miles per hour (mph). This sign means that you are not supposed to go faster than 25 mph, but there are many legal speeds you could drive, such as 22 mph, 24.5 mph or 19 mph. In a situation like this, which has more than one acceptable value, inequalities are used to represent the situation rather than equations. Solving multi-step inequalities is very similar to solving equations—what you do to one side you need to do to the other side in order to maintain the “balance” of the inequality. The Properties of Inequality can help you understand how to add, subtract, multiply, or divide within an inequality.

3A.1.1 Represent inequalities on a number line

First, let's define some important terminology. An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. Special symbols are used in these statements. When you read an inequality, read it from left to right—just like reading text on a page. In algebra, inequalities are used to describe large sets of solutions. Sometimes there are an infinite amount of numbers that will satisfy an inequality, so rather than try to list off an infinite amount of numbers, we have developed some ways to describe very large lists in succinct ways. The first way you are probably familiar with—the basic inequality. For example:
  • [latex]{x}\lt{9}[/latex] indicates the list of numbers that are less than 9. Would you rather write [latex]{x}\lt{9}[/latex] or try to list all the possible numbers that are less than 9? (hopefully, your answer is no)
  • [latex]-5\le{t}[/latex] indicates all the numbers that are greater than or equal to [latex]-5[/latex].
Note how placing the variable on the left or right of the inequality sign can change whether you are looking for greater than or less than. For example:
  • [latex]x\lt5[/latex] means all the real numbers that are less than 5, whereas;
  • [latex]5\lt{x}[/latex] means that 5 is less than x, or we could rewrite this with the x on the left: [latex]x\gt{5}[/latex] note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5, which is easier to interpret than 5 is less than x.
The second way is with a graph using the number line: A numberline. It is a long horizontal line with evenly spaced points, the middle of which is zero. And the third way is with an interval. We will explore the second and third ways in depth in this section. Again, those three ways to write solutions to inequalities are:
  • an inequality
  • an interval
  • a graph

Inequality Signs

The box below shows the symbol, meaning, and an example for each inequality sign. Sometimes it's easy to get tangled up in inequalities, just remember to read them from left to right.
Symbol Words Example
[latex]\neq [/latex] not equal to [latex]{2}\neq{8}[/latex], 2 is not equal to 8.
[latex]\gt[/latex] greater than [latex]{5}\gt{1}[/latex], 5 is greater than 1
[latex]\lt[/latex] less than [latex]{2}\lt{11}[/latex], 2 is less than 11
[latex] \geq [/latex] greater than or equal to [latex]{4}\geq{ 4}[/latex], 4 is greater than or equal to 4
[latex]\leq [/latex] less than or equal to [latex]{7}\leq{9}[/latex], 7 is less than or equal to 9
The inequality [latex]x>y[/latex] can also be written as [latex]{y}<{x}[/latex]. The sides of any inequality can be switched as long as the inequality symbol between them is also reversed.

3A.1.2 Graphing an Inequality

Inequalities can also be graphed on a number line. Below are three examples of inequalities and their graphs.  Graphs are a very helpful way to visualize information - especially when that information represents an infinite list of numbers! [latex]x\leq -4[/latex]. This translates to all the real numbers on a number line that are less than or equal to -4. Number line. Shaded circle on negative 4. Shaded line through all numbers less than negative 4. [latex]{x}\geq{-3}[/latex]. This translates to all the real numbers on the number line that are greater than or equal to -3. Number line. Shaded circle on negative 3. Shaded line through all numbers greater than negative 3. Each of these graphs begins with a circle—either an open or closed (shaded) circle. This point is often called the end point of the solution. A closed, or shaded, circle is used to represent the inequalities greater than or equal to [latex] \displaystyle \left(\geq\right) [/latex] or less than or equal to [latex] \displaystyle \left(\leq\right) [/latex]. The point is part of the solution. An open circle is used for greater than (>) or less than (<). The point is not part of the solution. The graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of [latex] \displaystyle x\geq -3[/latex] shown above, the end point is [latex]−3[/latex], represented with a closed circle since the inequality is greater than or equal to [latex]−3[/latex]. The blue line is drawn to the right on the number line because the values in this area are greater than [latex]−3[/latex]. The arrow at the end indicates that the solutions continue infinitely.

 Example 3A.1.A

Graph the inequality [latex]x\ge 4[/latex]

Answer: We can use a number line as shown. Because the values for x include 4, we place a solid dot on the number line at 4. Then we draw a line that begins at [latex]x=4[/latex] and, as indicated by the arrowhead, continues to positive infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4. A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.

This video shows an example of how to draw the graph of an inequality. https://youtu.be/-kiAeGbSe5c

Example 3A.1.B

Write and inequality describing all the real numbers on the number line that are less than 2, then draw the corresponding graph.

Answer: We need to start from the left and work right, so we start from negative infinity and end at [latex]-2[/latex]. We will not include either because infinity is not a number, and the inequality does not include [latex]-2[/latex]. Inequality: [latex]\left(-\infty,-2\right)[/latex] To draw the graph, place an open dot on the number line first, then draw a line extending to the left. Draw an arrow at the leftmost point of the line to indicate that it continues for infinity. Number line. Unshaded circle around 2 and shaded line through all numbers less than 2.

3A.1.3 Represent inequalities using interval notation

Another commonly used, and arguably the most concise, method for describing inequalities and solutions to inequalities is called interval notation. With this convention, sets are built with parentheses or brackets, each having a distinct meaning. The solutions to [latex]x\geq 4[/latex] are represented as [latex]\left[4,\infty \right)[/latex]. This method is widely used and will be present in other math courses you may take. The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be "equaled." A few examples of an interval, or a set of numbers in which a solution falls, are [latex]\left[-2,6\right)[/latex], or all numbers between [latex]-2[/latex] and [latex]6[/latex], including [latex]-2[/latex], but not including [latex]6[/latex]; [latex]\left(-1,0\right)[/latex], all real numbers between, but not including [latex]-1[/latex] and [latex]0[/latex]; and [latex]\left(-\infty,1\right][/latex], all real numbers less than and including [latex]1[/latex]. The table below outlines the possibilities. Remember to read inequalities from left to right, just like text. The table below describes all the possible inequalities that can occur and how to write them using interval notation, where a and b are real numbers.
Inequality Words Interval Notation
[latex]{a}\lt{x}\lt{ b}[/latex] all real numbers between a and b, not including a and b [latex]\left(a,b\right)[/latex]
[latex]{x}\gt{a}[/latex] All real numbers greater than a, but not including a [latex]\left(a,\infty \right)[/latex]
[latex]{x}\lt{b}[/latex] All real numbers less than b, but not including b [latex]\left(-\infty ,b\right)[/latex]
[latex]{x}\ge{a}[/latex] All real numbers greater than a, including a [latex]\left[a,\infty \right)[/latex]
[latex]{x}\le{b}[/latex] All real numbers less than b, including b [latex]\left(-\infty ,b\right][/latex]
[latex]{a}\le{x}\lt{ b}[/latex] All real numbers between a and b, including a [latex]\left[a,b\right)[/latex]
[latex]{a}\lt{x}\le{ b}[/latex] All real numbers between a and b, including b [latex]\left(a,b\right][/latex]
[latex]{a}\le{x}\le{ b}[/latex] All real numbers between a and b, including a and b [latex]\left[a,b\right][/latex]
[latex]{x}\lt{a}\text{ or }{x}\gt{ b}[/latex] All real numbers less than a or greater than b [latex]\left(-\infty ,a\right)\cup \left(b,\infty \right)[/latex]
All real numbers All real numbers [latex]\left(-\infty ,\infty \right)[/latex]

Example 3A.1.C

Describe the inequality [latex]x\ge 4[/latex] using interval notation

Answer: The solutions to [latex]x\ge 4[/latex] are represented as [latex]\left[4,\infty \right)[/latex]. Note the use of a bracket on the left because 4 is included in the solution set.

Example 3A.1.D

Use interval notation to indicate all real numbers greater than or equal to [latex]-2[/latex].

Answer: Use a bracket on the left of [latex]-2[/latex] and parentheses after infinity: [latex]\left[-2,\infty \right)[/latex]. The bracket indicates that [latex]-2[/latex] is included in the set with all real numbers greater than [latex]-2[/latex] to infinity.

Think About It

In the previous examples you were given an inequality or a description of one with words and asked to draw the corresponding graph and write the interval. In this example you are given an interval and asked to write the inequality and draw the graph. Given [latex]\left(-\infty,10\right)[/latex], write the associated inequality and draw the graph. In the box below, write down whether you think it will be easier to draw the graph first or write the inequality first. [practice-area rows="1"][/practice-area]

Answer: We will draw the graph first. The interval reads "all real numbers less than 10," so we will start by placing an open dot on 10 and drawing a line to the left with an arrow indicating the solution continues to negative infinity. An open circle on 10 and a line going from 10 to all numbers below 10. To write the inequality, we will use < since the parentheses indicate that 10 is not included. [latex]x<10[/latex]

In the following video, you will see examples of how to write inequalities in the three ways presented here: as an inequality, in interval notation, and with a graph. https://youtu.be/X0xrHKgbDT0

3A.1.4 Multiplication and Division Properties of Inequality

Solving inequalities is very similar to solving equations, except you have to reverse the inequality symbols when you multiply or divide both sides of an inequality by a negative number. There are three ways to represent solutions to inequalities: an interval, a graph, and an inequality. Because there is usually more than one solution to an inequality, when you check your answer you should check the end point and one other value to check the direction of the inequality.When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol. The following table illustrates how the multiplication property is applied to inequalities, and how multiplication by a negative reverses the inequality:
Start With Multiply By Final Inequality
[latex]a>b[/latex] [latex]c[/latex] [latex]ac>bc[/latex]
[latex]5>3[/latex] [latex]3[/latex] [latex]15>9[/latex]
[latex]a>b[/latex] [latex]-c[/latex] [latex]-ac<-bc[/latex]
[latex]5>3[/latex] [latex]-3[/latex] [latex]-15<-9[/latex]
The following table illustrates how the division property is applied to inequalities, and how dividing by a negative reverses the inequality:
Start With Divide By Final Inequality
[latex]a>b[/latex] [latex]c[/latex] [latex] \displaystyle \frac{a}{c}>\frac{b}{c}[/latex]
[latex]4>2[/latex] [latex]2[/latex] [latex] \displaystyle \frac{4}{2}>\frac{2}{2}[/latex]
[latex]a>b[/latex] [latex]-c[/latex] [latex] \displaystyle -\frac{a}{c}<-\frac{b}{c}[/latex]
[latex]4>2[/latex] [latex]-2[/latex] [latex] \displaystyle -\frac{4}{2}<-\frac{2}{2}[/latex]
In the first example, we will show how to apply the multiplication and division properties of equality to solve some inequalities.

Example 3A.1.E

Illustrate the multiplication property for inequalities by solving each of the following:
  1. [latex]3x<6[/latex]
  2. [latex]-2x - 1\ge 5[/latex]
  3. [latex]5-x>10[/latex]

Answer:

  1. [latex]\begin{array}{l}3x<6\hfill \\ \frac{1}{3}\left(3x\right)<\left(6\right)\frac{1}{3}\hfill \\ x<2\hfill \end{array}[/latex]
  2. [latex]\begin{array}{ll}-2x - 1\ge 5\hfill & \hfill \\ -2x\ge 6\hfill & \hfill \\ \left(-\frac{1}{2}\right)\left(-2x\right)\ge \left(6\right)\left(-\frac{1}{2}\right)\hfill & \text{Multiply by }-\frac{1}{2}.\hfill \\ x\le -3\hfill & \text{Reverse the inequality}.\hfill \end{array}[/latex]
  3. [latex]\begin{array}{ll}5-x>10\hfill & \hfill \\ -x>5\hfill & \hfill \\ \left(-1\right)\left(-x\right)>\left(5\right)\left(-1\right)\hfill & \text{Multiply by }-1.\hfill \\ x<-5\hfill & \text{Reverse the inequality}.\hfill \end{array}[/latex]

3A.1.5 Solve Inequalities Using the Addition Property

When we solve equations we may need to add or subtract in order to isolate the variable, the same is true for inequalities. There are no special behaviors to watch out for when using the addition property to solve inequalities. The following table illustrates how the addition property applies to inequalities.
Start With Add Final Inequality
[latex]a>b[/latex] [latex]c[/latex] [latex]a+c>b+c[/latex]
[latex]5>3[/latex] [latex]3[/latex] [latex]8>6[/latex]
[latex]a>b[/latex] [latex]-c[/latex] [latex]a-c>b-c[/latex]
[latex]5>3[/latex] [latex]-3[/latex] [latex]2>0[/latex]
These properties also apply to [latex]a\le b[/latex], [latex]a>b[/latex], and [latex]a\ge b[/latex]. In our next example we will use the addition property to solve inequalities.

Example 3A.1.F

Illustrate the addition property for inequalities by solving each of the following:
  1. [latex]x - 15<4[/latex]
  2. [latex]6\ge x - 1[/latex]
  3. [latex]x+7>9[/latex]

Answer: The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.

  1. [latex]\begin{array}{ll}x - 15<4\hfill & \hfill \\ x - 15+15<4+15 \hfill & \text{Add 15 to both sides.}\hfill \\ x<19\hfill & \hfill \end{array}[/latex]
  2. [latex]\begin{array}{ll}6\ge x - 1\hfill & \hfill \\ 6+1\ge x - 1+1\hfill & \text{Add 1 to both sides}.\hfill \\ 7\ge x\hfill & \hfill \end{array}[/latex]
  3. [latex]\begin{array}{ll}x+7>9\hfill & \hfill \\ x+7 - 7>9 - 7\hfill & \text{Subtract 7 from both sides}.\hfill \\ x>2\hfill & \hfill \end{array}[/latex]

The following video shows examples of solving single-step inequalities using the multiplication and addition properties. https://youtu.be/1Z22Xh66VFM The following video show examples of solving inequalities with the variable on the right side. https://youtu.be/RBonYKvTCLU

3A.1.6 Solve Multi-Step Inequalities

As the previous examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations. To isolate the variable and solve, we combine like terms and perform operations with the multiplication and addition properties.

Example 3A.1.G

Solve the inequality: [latex]13 - 7x\ge 10x - 4[/latex].

Answer: Solving this inequality is similar to solving an equation up until the last step.

[latex]\begin{array}{ll}13 - 7x\ge 10x - 4\hfill & \hfill \\ 13 - 17x\ge -4\hfill & \text{Move variable terms to one side of the inequality}.\hfill \\ -17x\ge -17\hfill & \text{Isolate the variable term}.\hfill \\ x\le 1\hfill & \text{Dividing both sides by }-17\text{ reverses the inequality}.\hfill \end{array}[/latex]
The solution set is given by the interval [latex]\left(-\infty ,1\right][/latex], or all real numbers less than and including 1.

In the next example we solve an inequality that contains fractions, not how we need to reverse the inequality sign at the end because we multiply by a negative.

Example 3A.1.H

Solve the following inequality and write the answer in interval notation: [latex]-\frac{3}{4}x\ge -\frac{5}{8}+\frac{2}{3}x[/latex].

Answer: We begin solving in the same way we do when solving an equation.

[latex]\begin{array}{ll}-\frac{3}{4}x\ge -\frac{5}{8}+\frac{2}{3}x\hfill & \hfill \\ -\frac{3}{4}x-\frac{2}{3}x\ge -\frac{5}{8}\hfill & \text{Put variable terms on one side}.\hfill \\ -\frac{9}{12}x-\frac{8}{12}x\ge -\frac{5}{8}\hfill & \text{Write fractions with common denominator}.\hfill \\ -\frac{17}{12}x\ge -\frac{5}{8}\hfill & \hfill \\ x\le -\frac{5}{8}\left(-\frac{12}{17}\right)\hfill & \text{Multiplying by a negative number reverses the inequality}.\hfill \\ x\le \frac{15}{34}\hfill & \hfill \end{array}[/latex]
The solution set is the interval [latex]\left(-\infty ,\frac{15}{34}\right][/latex].

Licenses & Attributions

CC licensed content, Original

  • Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
  • Revision and Adaptation. License: CC BY: Attribution.
  • Graph Linear Inequalities in One Variable (Basic). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.

CC licensed content, Shared previously

  • Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
  • Ex: Graph Basic Inequalities and Express Using Interval Notation. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
  • College Algebra. Provided by: Lumen Learning Authored by: Jay Abramson, et al.. License: CC BY: Attribution.
  • Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Left Side). Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
  • Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Right Side). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Ex: Solve One Step Linear Inequality by Dividing (Variable Left). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Ex: Solve One Step Linear Inequality by Dividing (Variable Right). Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.