6.3.3: Efficiently Solving Inequalities

Lesson

Let's solve more complicated inequalities.

Exercise \(\PageIndex{1}\): Lots of Negatives

Here is an inequality: \(-x\geq -4\).

1. Predict what you think the solutions on the number line will look like.
2. Select all the values that are solutions to \(-x\geq -4\):
   1. \(3\)
   2. \(-3\)
   3. \(4\)
   4. \(-4\)
   5. \(4.001\)
   6. \(-4.001\)
3. Graph the solutions to the inequality on the number line:

Figure \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\): Inequalities with Tables

1. Let's investigate the inequality \(x-3>-2\).

Table \(\PageIndex{1}\)

\(x\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(x-3\)	\(-7\)		\(-5\)				\(-1\)		\(1\)

1. Complete the table.
2. For which values of \(x\) is it true that \(x-3=-2\)?
3. For which values of \(x\) is it true that \(x-3>-2\)?
4. Graph the solutions to \(x-3>-2\) on the number line:

Figure \(\PageIndex{2}\)

2. Here is an inequality: \(2x<6\).

1. Predict which values of \(x\) will make the inequality \(2x<6\) true.
2. Complete the table. Does it match your prediction?

   Table \(\PageIndex{2}\)

   \(x\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
   \(2x\)

3. Graph the solutions to \(2x<6\) on the number line:

Figure \(\PageIndex{3}\)

3. Here is an inequality: \(-2x<6\).

1. Predict which values of \(x\) will make the inequality \(-2x<6\) true.
2. Complete the table. Does it match your prediction?

   Table \(\PageIndex{4}\)

   \(x\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
   \(-2x\)

3. Graph the solutions to \(-2x<6\) on the number line:

Figure \(\PageIndex{4}\)

d. How are the solutions to \(2x<6\) different from the solutions to \(-2x<6\)?

Exercise \(\PageIndex{3}\): Which Side are the Solutions?

1. Let's investigate \(-4x+5\geq 25\).

1. Solve \(-4x+5=25\).
2. Is \(-4x+5\geq 25\) true when \(x\) is 0? What about when \(x\) is \(7\)? What about when \(x\) is \(-7\)?
3. Graph the solutions to \(-4x+5\geq 25\) on the number line.

Figure \(\PageIndex{5}\)

2. Let's investigate \(\frac{4}{3}x+3<\frac{23}{3}\).

1. Solve \(\frac{4}{3}x+3=\frac{23}{3}\).
2. Is \(\frac{4}{3}x+3<\frac{23}{3}\) true when \(x\) is \(0\)?
3. Graph the solutions to \(\frac{4}{3}x+3<\frac{23}{3}\) on the number line.

Figure \(\PageIndex{6}\)

3. Solve the inequality \(3(x+4)>17.4\) and graph the solutions on the number line.

Figure \(\PageIndex{7}\)

4. Solve the inequality \(-3\left(x-\frac{4}{3}\right)\leq 6\) and graph the solutions on the number line.

Figure \(\PageIndex{8}\)

Are you ready for more?

Write at least three different inequalities whose solution is \(x>-10\). Find one with \(x\) on the left side that uses a \(<\).

Summary

Here is an inequality: \(3(10-2x)<18\). 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 \(3(10-2x)=18\) to get the solution \(x=2\). That means 2 is the boundary between values of 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 \(3(10-2\cdot 5)<18\) 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:

Figure \(\PageIndex{9}\)

Notice that 2 itself is not a solution because it's the value of \(x\) that makes \(3(10-2x)\) ​equal to 18, and so it does not make \(3(10-2x)<18\) 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 \(3(10-2\cdot 0)<18\) 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

Definition: 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.

Practice

Exercise \(\PageIndex{4}\)

1. Consider the inequality \(-1\leq \frac{x}{2}\).
   1. Predict which values of \(x\) will make the inequality true.
   2. Complete the table to check your prediction.

      Table \(\PageIndex{5}\)

      \(x\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
      \(\frac{x}{2}\)

2. Consider the inequality \(1\leq\frac{-x}{2}\).
   1. Predict which values of \(x\) will make it true.
   2. Complete the table to check your prediction.

      Table \(\PageIndex{6}\)

      \(x\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
      \(-\frac{x}{2}\)

Exercise \(\PageIndex{5}\)

Diego is solving the inequality \(100-3x\geq -50\). He solves the equation \(100-3x=-50\) and gets \(x=50\). What is the solution to the inequality?

1. \(x<50\)
2. \(x\leq 50\)
3. \(x>50\)
4. \(x\geq 50\)

Exercise \(\PageIndex{6}\)

Solve the inequality \(-5(x-1)>-40\), and graph the solution on a number line.

Exercise \(\PageIndex{7}\)

Select all values of \(x\) that make the inequality \(-x+6\geq 10\) true.

1. \(-3.9\)
2. \(4\)
3. \(-4.01\)
4. \(-4\)
5. \(4.01\)
6. \(3.9\)
7. \(0\)
8. \(-7\)

(From Unit 6.3.1)

Exercise \(\PageIndex{8}\)

Draw the solution set for each of the following inequalities.

1. \(x>7\)

Figure \(\PageIndex{10}\)

2. \(x\geq -4.2\)

Figure \(\PageIndex{11}\)

(From Unit 6.3.1)

Exercise \(\PageIndex{9}\)

The price of a pair of earrings is $22 but Priya buys them on sale for $13.20.

1. By how much was the price discounted?
2. What was the percentage of the discount?

(From Unit 4.3.3)