We have now looked at solving inequalities using one or two operations at a time. In particular, we have seen 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.
Remember 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.
Exploration
Let's now take a look at solving an inequality that involves more than one variable term, such as $8x-5<5x+7$8x−5<5x+7. There are two terms involving $x$x, with one on either side of the inequality symbol.
To go about solving this inequality, we first want to group these variable terms together. We can do so by subtracting $5x$5x from both sides, which leaves us with the inequality $3x-5<7$3x−5<7. This looks like a very familiar two-step inequality! We can solve it as we have done before, by using the order of operations in reverse.
| $8x-5$8x−5 | $<$< | $5x+7$5x+7 | ||
| $3x-5$3x−5 | $<$< | $7$7 | Subtracting $5x$5x from both sides | |
| $3x$3x | $<$< | $12$12 | Adding $5$5 to both sides | |
| $x$x | $<$< | $4$4 | Dividing both sides by $3$3 |
In this case, we arrive at the result $x<4$x<4.
Now, we didn't have to start by subtracting $5x$5x from both sides - we could have instead subtracted $8x$8x from both sides to group the variable terms on the right instead of the left. Let's have a look at this path:
| $8x-5$8x−5 | $<$< | $5x+7$5x+7 | ||
| $-5$−5 | $<$< | $-3x+7$−3x+7 | Subtracting $8x$8x from both sides | |
| $-12$−12 | $<$< | $-3x$−3x | Subtracting $7$7 from both sides | |
| $4$4 | $>$> | $x$x | Dividing both sides by $-3$−3 (and reversing the inequality symbol) |
Once again, we arrive at the result $x<4$x<4. Notice, however, that this solution path involved dividing by a negative number at one point.
For an inequality of this form, we can usually choose to avoid dividing by a negative number by paying attention to the coefficients. Notice that in the first path we subtracted $5x$5x - the term with the smaller coefficient - from both sides of the inequality. This left a positive variable term, and so we didn't have to divide by a negative at any point.
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.
- If there is more than one variable term, start by grouping the variable terms. In particular, moving the term with the smaller coefficient usually leads to a simpler solution path.
Practice questions
Question 1
Solve the following inequality: $4\left(2x+3\right)>-4$4(2x+3)>−4
Question 2
Solve the following inequality: $\frac{3-2x}{5}<3$3−2x5<3
Question 3
Solve the following inequality: $7x+4>x+16$7x+4>x+16