4.11 Two-step inequalities

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.

 

Exploration

Let's take a look at solving a slightly more complicated inequality, such as $-3x+2\ge14$−3x+2≥14. There are now two operations being applied to $x$x (multiplication and addition). Much like solving equations with two (or more) operations, we will need to take the order of operations into consideration as well.

Looking at the inequality $-3x+2\ge14$−3x+2≥14 and thinking about the order of operations, we can see that $x$x is first multiplied by $-3$−3 and then $2$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$2 from both sides, then dividing both sides by $-3$−3 (which will change the inequality symbol used):

$-3x+2$−3x+2	$\ge$≥	$14$14	Given
$-3x+2-2$−3x+2−2	$\ge$≥	$14-2$14−2	Subtracting $2$2 from both sides
$-3x$−3x	$\ge$≥	$12$12	Simplifying
$\frac{-3x}{-3}$−3x−3​	$\le$≤	$\frac{12}{-3}$12−3​	Dividing both sides by $-3$−3 (and changing the inequality symbol)
$x$x	$\le$≤	$-4$−4	Simplifying

In this case, we arrive at the result $x\le-4$x≤−4. We can test some values in the original inequality to see if this is the right solution set - let's say $x=-5$x=−5 and $x=-3$x=−3.

• When $x=-5$x=−5, we have $-3x+2=-3\times\left(-5\right)+2=17$−3x+2=−3×(−5)+2=17, which is greater than or equal to $14$14.
• When $x=-3$x=−3, we have $-3x+2=-3\times\left(-3\right)+2=11$−3x+2=−3×(−3)+2=11, which is not greater than or equal to $14$14.

So our result of $x\le-4$x≤−4 seems to be correct. We can graph this on the number line if required.

 

 

Practice questions

Question 1

Question 2

Question 3