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Solving One-Step Inequalities

Solving an inequality means finding the value or values that make the inequality true. In other words, finding the solution set. This can be done, similar to solving equations, by using inverse operations.
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Exercise

Solve the following inequality and graph its solution set. x+37 x+3\leq 7

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Solution
To solve the inequality, we must isolate x.x. Notice that the inverse of adding 33 is subtracting 3.3.
x+37x+3\leq 7
x+3373x+3-3\leq 7-3
x4x\leq 4
The inequality has the solutions x4.x\leq 4. To graph this on a number line, we can place a closed circle at 4,4, because xx can equal 4.4. Additionally, since xx can take all values less than 4,4, we shade the region to the left.
Rule

Multiplying and Dividing Inequalities with a Negative

There is one important difference between solving equations and solving inequalities. When dividing or multiplying the inequality by a negative number, the inequality symbol must be reversed or flipped.

Consider the following example to understand why the inequality symbol is reversed. The inequality 2 <  5 2 \ < \ \ 5 is a true statement. Multiplying both sides by -1\text{-}1 results in -2  >  -5. \text{-} 2 \ \ \fbox{\phantom{>}} \ \ \text{-} 5.

Notice that -5\text{-} 5 is smaller than -2.\text{-} 2. Thus, the inequality symbol that makes this a true statement is >.>. Therefore, when multiplying or dividing an inequality by a negative number, the inequality symbol must be reversed.
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Exercise

Solve the following inequality and graph its solution set. -2x>6 \text{-} 2x>6

Show Solution
Solution
To isolate xx we can divide both sides of the inequality by -2.\text{-} 2.
-2x>6\text{-} 2x>6
-2x-2<6-2\dfrac{\text{-} 2x}{\text{-} 2}<\dfrac{6}{\text{-} 2}
x<-3x<\text{-} 3
The solutions x<-3x< \text{-} 3 satisfy the inequality. We can graph the solution set on a number line. Since xx cannot equal -3,\text{-} 3, we place an open circle at -3.\text{-} 3. Additionally, to show all values less than -3,\text{-} 3, we'll shade the region to the left.
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