Lesson 15
Efficiently Solving Inequalities
Let’s solve more complicated inequalities.
15.1: Lots of Negatives
Here is an inequality: \(\textx \geq \text4\).
 Predict what you think the solutions on the number line will look like.
 Select all the values that are solutions to \(\textx \geq \text4\):
 3
 3
 4
 4
 4.001
 4.001
 Graph the solutions to the inequality on the number line:
15.2: Inequalities with Tables

Let's investigate the inequality \(x3>\text2\).
\(x\) 4 3 2 1 0 1 2 3 4 \(x3\) 7 5 1 1  Complete the table.
 For which values of \(x\) is it true that \(x  3 = \text2\)?
 For which values of \(x\) is it true that \(x  3 > \text2\)?
 Graph the solutions to \(x  3 > \text2\) on the number line:

Here is an inequality: \(2x<6\).
 Predict which values of \(x\) will make the inequality \(2x < 6\) true.

Complete the table. Does it match your prediction?
\(x\) 4 3 2 1 0 1 2 3 4 \(2x\) 
Graph the solutions to \(2x < 6\) on the number line:

Here is an inequality: \(\text2x<6\).
 Predict which values of \(x\) will make the inequality \(\text2x < 6\) true.

Complete the table. Does it match your prediction?
\(x\) 4 3 2 1 0 1 2 3 4 \(\text2x\)  Graph the solutions to \(\text2x < 6\) on the number line:
 How are the solutions to \(2x<6\) different from the solutions to \(\text2x<6\)?
15.3: Which Side are the Solutions?
 Let’s investigate \(\text4x + 5 \geq 25\).
 Solve \(\text4x+5 = 25\).
 Is \(\text4x + 5 \geq 25\) true when \(x\) is 0? What about when \(x\) is 7? What about when \(x\) is 7?
 Graph the solutions to \(\text4x + 5 \geq 25\) on the number line.
 Let's investigate \(\frac{4}{3}x+3 < \frac{23}{3}\).
 Solve \(\frac43x+3 = \frac{23}{3}\).
 Is \(\frac{4}{3}x+3 < \frac{23}{3}\) true when \(x\) is 0?

Graph the solutions to \(\frac{4}{3}x+3 < \frac{23}{3}\) on the number line.
 Solve the inequality \(3(x+4) > 17.4\) and graph the solutions on the number line.
 Solve the inequality \(\text3\left(x\frac43\right) \leq 6\) and graph the solutions on the number line.
Write at least three different inequalities whose solution is \(x > \text10\). Find one with \(x\) on the left side that uses a \(<\).
Summary
Here is an inequality: \(3(102x) < 18\). The solution to this inequality is all the values you could use in place of \(x\) to make the inequality true.
In order to solve this, we can first solve the related equation \(3(102x) = 18\) to get the solution \(x = 2\). That means 2 is the boundary between values of \(x\) that make the inequality true and values that make the inequality false.
To solve the inequality, we can check numbers greater than 2 and less than 2 and see which ones make the inequality true.
Let’s check a number that is greater than 2: \(x= 5\). Replacing \(x\) with 5 in the inequality, we get \(3(102 \boldcdot 5) < 18\) or just \(0 < 18\). This is true, so \(x=5\) is a solution. This means that all values greater than 2 make the inequality true. We can write the solutions as \(x > 2\) and also represent the solutions on a number line:
Notice that 2 itself is not a solution because it's the value of \(x\) that makes \(3(102x)\) equal to 18, and so it does not make \(3(102x) < 18\) true.
For confirmation that we found the correct solution, we can also test a value that is less than 2. If we test \(x=0\), we get \(3(102 \boldcdot 0) < 18\) or just \(30 < 18\). This is false, so \(x = 0\) and all values of \(x\) that are less than 2 are not solutions.
Glossary Entries
 solution to an inequality
A solution to an inequality is a number that can be used in place of the variable to make the inequality true.
For example, 5 is a solution to the inequality \(c<10\), because it is true that \(5<10\). Some other solutions to this inequality are 9.9, 0, and 4.