Print bookPrint book

REVIEW: Solving Linear Inequalities

Solving Linear Inequalities

Site: Mountain Heights Academy OpenCourseWare
Course: Mathematics Essentials Q4 v2013
Book: REVIEW: Solving Linear Inequalities
Printed by: Guest user
Date: Saturday, 18 November 2017, 8:46 AM

WATCH: Solving Inequalities

Previous Material: Solving Linear Inequalities Part 1

The material in each of the following subsections is the same material you were given in Quarter 3 on Solving Linear Inequalities. Please feel free to review them as needed.

REVIEW LESSON WATCH: Linear Inequalities

WATCH: Modeling Inequalities

EXAMPLE 1

EXAMPLE 2

EXAMPLE 3

EXAMPLE 4

READ: Graph Inequalities on a Number Line

Graph Inequalities on a Number Line

In the last few lessons you have been learning about equations and about balancing an equation. Let’s think about equations. An equation is a number statement with an equal sign. The equal sign tells us that the quantity on one side of the equation is equal the quantity on the other side of the equation. We can solve an equation by figuring out the quantity that will make the equation a true statement.

Example 1

x+5=12

If we think about this equation, we can use mental math and we know that the unknown quantity is equal to 7. If we substitute 7 in for x, we will have a true statement.

7 + 5 &= 12\\ 12 &= 12

Our equation is balanced because one side is equal to the other side. Notice that there is one answer for x that makes this a true statement.


What is an inequality?

An inequality is a mathematical statement that can be equal, but can also be unequal.

We use the following symbols to show that we are working with an inequality.

> means greater than

< means less than

\ge means greater than or equal to

\le means less than or equal to


How do we apply these symbols?

Well, if you think about it, we use the symbols to make a true statement. Let’s look at an example.


Example 2

x + 3 > 5

There are many possible answers that will make this a true statement. We need the quantity on the left side of the inequality to create a sum that is greater than five. Notice that the sign does not have a line under it. We want a quantity that is greater than five not greater than or equal to five on the left side of the inequality.

To make this true, we can choose a set of numbers that has a 3 or greater in it.

x = \{3, 4, 5 \ldots .\}


We don’t need to worry about solving inequalities yet, the key thing to notice is that there are many possible answers that will make an inequality a true statement.

We can use graphs to help us understand inequalities in a visual way.

Graphing inequalities on a number line can help us understand which numbers are s solutions for the inequality and which numbers are not solutions.

Here are some tips for graphing inequalities on a number line.

  • Use an open circle to show that a value is not a solution for the inequality. You will use open circles to graph inequalities that include the symbols > or < .
  • Use a closed circle to show that a value is a solution for the inequality. You will use closed circles to graph inequalities that include the symbols \ge or \le.

Example 3

Graph this inequality x > 3

To complete this task, first draw a number line from –5 to 5.

The inequality x > 3 is read as “x is greater than 3.” So, the solutions of this inequality include all numbers greater than 3. It does not include 3, so draw an open circle at 3 to show that 3 is not a solution for this inequality. Then draw an arrow showing all numbers greater than 3. The arrow should face right because the greater numbers are to the right on a number line.

The graph above shows the solutions for the inequality x > 3.


The graph above can help you see which numbers are solutions for x > 3 and which are not. For example, the arrow includes the numbers 3.5, 4, and 5. If you continued the number line, the arrow would also include the numbers 10 and 100. So, all those numbers––3.5, 4, 5, 10, and 100––are possible values for x.

Example 4

Graph this inequality x<-1.

First, draw a number line from –5 to 5.

The inequality x<-1 is read as “x is less than -1.” So, the solutions of this inequality include all numbers less than -1. It does not include -1, so draw an open circle at -1 to show that -1 is not a solution for this inequality. Then draw an arrow showing all numbers less than -1. The arrow should face left because the lesser numbers are to the left on a number line.

The graph above shows the solutions for the inequality x<-1.


Example 5

Graph this inequality x \ge 0.

First, draw a number line from –5 to 5.

The inequality x \ge 0 is read as “x is greater than or equal to 0.” So, the solutions of this inequality include zero and all numbers that are greater than 0. Draw a closed circle at 0 to show that 0 is a solution for this inequality. Then draw an arrow showing all numbers greater than 0.

The graph above shows the solutions for the inequality x \ge 0.

EXAMPLE 5

EXAMPLE 6

CHECK Yourself! Understand Inequalities

READ: Recognize Equivalent Inequalities

Recognize Equivalent Inequalities

Sometimes, you may need to rewrite an inequality as an equivalent inequality in order to better understand it. This means that an inequality can be written in two different ways so that we can understand it. Look at the example below.


Example 1

Graph this inequality 4 \ge x.

Draw a number line from –5 to 5.

The inequality 4 \ge x is read as “4 is greater than or equal to x.” This inequality will be easier to understand if we rewrite it so that the variable is listed first. If we list the x first, we must reverse the inequality symbol. That means changing the “greater than or equal to” symbol (\ge) to a “less than or equal to symbol” \le.

4 \ge x is equivalent to x \le 4.

This makes sense. If 4 is greater than or equal to x, then x must be less than or equal to 4.

The inequality x \le 4 is read as “x is less than or equal to 4.” So, the solutions of this inequality include 4 and all numbers that are less than 4. Draw a closed circle at 4 to show that 4 is a solution for this inequality. Then draw an arrow showing all numbers less than 4.

The graph above shows the solutions for the inequality x \le 4.


Example 2

Write an equivalent inequality for x \ge 5.

To write an equivalent inequality, we reverse the terms. If x is greater than or equal to five then five is less than or equal to x. Let’s rewrite this.

The equivalent inequalities are x \ge 5and 5 \le x.

CHECK Yourself! Inequalities

READ: Solve Inequalities and Graph Solutions

Solve Inequalities and Graph Solutions

Now that you have developed an understanding of inequalities we can solve them and graph the solution sets.

We can solve an inequality in a similar way as we would use to solve an equation. The tricky part is in the answer not in the process. Once we solve the inequality, we can graph the solution. Let’s look at an example.


Example 1

Solve this inequality and graph its solution n-4 \le 3.

Solve the inequality as you would solve an equation, by using inverse operations. Since the 4 is subtracted from n, add 4 to both sides of the inequality to solve it. You do not need to multiply or divide both sides by a negative number, so you do not need to reverse the inequality symbol. The symbol should stay the same.

n-4 & \le 3\\ n-4 +4 & \le 3+4\\ n+(-4+4) & \le 7\\ n+0 & \le 7\\ n & \le 7

Now, graph the solution. The inequality n \le 7 is read as “n is less than or equal to 7.” So, the solutions of this inequality include 7 and all numbers that are less than 7.

Draw a number line from 0 to 10. Add a closed circle at 7 to show that 7 is a solution for this inequality. Then draw an arrow showing all numbers less than 7.

The solution for this inequality is n \le 7, and its graph is shown above.


Example 2

Solve this inequality and graph its solution -2n<14.

Solve this inequality as you would solve an equation, by using inverse operations. Since the is multiplied by the n, divide both sides of the inequality by to solve it. Since this involves multiplying both sides of the inequality by a negative number, the sense of the inequality will change and you will need to reverse the inequality symbol. This means changing the inequality symbol from a “less than” symbol (<) to a “greater than” symbol (>).

-2n & < 14\\ \frac{-2n}{-2} & > \frac{14}{-2}\\ 1n & > -7\\ n & > -7

Now, graph the solution. The inequality n>-7 is read as “n is greater than or equal to 7.” So, the solutions of this inequality include all numbers that are greater than 7.

Draw a number line from –10 to 0. Add an open circle at -7 to show that -7 is not a solution for this inequality. Then draw an arrow showing all numbers greater than -7.

The solution for this inequality is n>-7, and its graph is shown above.


Sometimes, you will need to take more than one step to solve an inequality. You can think of these problems in the same way that you thought about two-step equations.


Example 3

\frac{n}{3}+9 \ge -9.

Solve this inequality as you would solve an equation, by using inverse operations. First, try to get the term with the variable, \frac{n}{3}, by itself on one side of the inequality. Since the 9 is being added to \frac{n}{3}, subtract 9 from both sides of the inequality. You do not need to multiply or divide both sides by a negative number, so you do not need to reverse the inequality symbol during this step.

\frac{n}{3}+9 & \ge -9\\ \frac{n}{3}+9-9 & \ge -9-9\\ \frac{n}{3}+0 & \ge (-9+-9)\\ \frac{n}{3} & \ge -18

There is a second step you must take to find the solution. Since n is divided by 3, you must multiply both sides of the inequality by 3 to find its solution. This involves multiplying by a positive number, 3, so you do not need to reverse the inequality symbol. Be careful! It is true that you will need to multiply 3 by –18 to find the solution. However, since you are not multiplying both sides of the inequality by a negative number, you do not reverse the inequality symbol.

\frac{n}{3} & \ge -18\\ \frac{n}{3} \times 3 & \ge -18 \times 3\\ \frac{n}{3} \times \frac{3}{1} & \ge -54\\ \frac{n}{\cancel{3}} \times \frac{\cancel{3}}{1} & \ge -54\\ \frac{n}{1} & \ge -54\\ n & \ge -54

The solution for this inequality is n \ge -54.

CHECK Yourself! Graphing Inequalities

Previous Material: Solving Linear Inequalities Part 2

The material in each of the following subsections is the same material you were given in Quarter 3 on Solving Linear Inequalities. Please feel free to review them as needed.

WATCH: Solving & Graphing Inequalities using Addition & Subtraction

EXAMPLE 1

EXAMPLE 2

EXAMPLE 3

EXAMPLE 4

EXAMPLE 5

EXAMPLE 6

CHECK Yourself! Solving Inequalities with Addition and Subtraction

WATCH: Solving & Graphing Inequalities using Multiplication & Division

EXAMPLE 1

EXAMPLE 2

EXAMPLE 3

EXAMPLE 4

EXAMPLE 5

EXAMPLE 6

EXAMPLE 7

CHECK Yourself! Solving Inequalities using Multiplication or Division