6.07 Two-step inequalities
Introduction
We have looked at solving one step inequalities. We learned that the process is almost identical to that of solving equations, but we also need to keep in mind which operations cause the inequality symbol to reverse.
In particular, we found that multiplying or dividing by a negative number causes the inequality symbol to change direction. Also, writing an inequality in reverse order causes the inequality symbol to reverse.
Two-step inequalities
Let's take a look at solving a two-step inequality, such as -3x+2 \geq 14. There are now two operations being applied to x (multiplication and addition). Just like when we were solving equations with two (or more) operations, we will need to consider order of operations here as well.
Looking at the inequality -3x+2 \geq 14 and thinking about the order of operations, we can see that x is first multiplied by -3 and then 2 is added. To solve this inequality, we want to undo these operations in reverse order. That is, we can solve this inequality by first subtracting 2 from both sides, then dividing both sides by -3 (which will change the inequality symbol used):
| \displaystyle -3x+2 | \displaystyle \geq | \displaystyle 14 | |
| \displaystyle -3x+2-2 | \displaystyle \geq | \displaystyle 14-2 | Subtract 2 from both sides |
| \displaystyle -3x | \displaystyle \geq | \displaystyle 12 | Simplify |
| \displaystyle \dfrac{-3x}{-3} | \displaystyle \leq | \displaystyle \dfrac{12}{-3} | Divide both sides by -3, reverse the inequality symbol |
| \displaystyle x | \displaystyle \leq | \displaystyle -4 | Simplify |
We found that x\leq-4. We can test some values in the given inequality to see if this is the solution set is true. Let's try the numbers just above and below -4, say x=-5 and x=-3.
- When x=-5, we have -3x+2 = -3 \times (-5) +2 = 17, which is greater than or equal to 14.
- When x=-3, we have -3x+2 = -3 \times (-3) +2 = 11, which is not greater than or equal to 14.
So our result of x\leq-4 seems to be correct. We can also graph this on the number line. For x\leq-4, we will include -4 on the number line as a filled circle and a ray pointing to the left side.
When solving any inequality:
- Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.
- Writing an inequality in reverse order also reverses the inequality symbol.
When solving an inequality with two (or more) operations:
- It is usually easiest to undo one operation at a time, in reverse order to the order of operations.
Examples
Example 1
Solve the following inequality: 3x+27>3
Example 2
Solve the following inequality: \dfrac{a}{5} + 3 > 3
Example 3
Consider the inequality 7-x>13.
Solve the inequality.
Now, plot the solutions to the inequality 7-x>13 on a number line.
When solving any inequality:
- Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.
- Writing an inequality in reverse order also reverses the inequality symbol.
When solving an inequality with two (or more) operations:
- It is generally easiest to undo one operation at a time, in reverse order to the order of operations.