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2.08 Multistep inequalities

Lesson

Solve three step inequalities

We have now looked at solving inequalities using one operation 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$8x5<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$3x5<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$8x5 $<$< $5x+7$5x+7  
$3x-5$3x5 $<$< $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$8x5 $<$< $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.

 

Summary

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$32x5<3

Question 3

Solve the following inequality: $7x+4>x+16$7x+4>x+16

 

Inequalities on a number line

In the previous examples we solved, but were not required to graph the solution to the inequality on a number line.  When we wish to graph the solution set, it is helpful for us to remember how to read each of the inequality symbols.  There are four main types of inequalities. Here are examples of each type.

$x<2$x<2 "$x$x is less than $2$2"
$x>-5$x>5 "$x$x is greater than $-5$5"
$x\le-4$x4 "$x$x is less than or equal to $-4$4"
$x\ge17$x17 "$x$x is greater than or equal to $17$17"

 

Inequalities on the number line

Now, what if we wanted to plot an inequality, such as $x\le4$x4?

When you say "$x$x less than or equal to $4$4", you're not just talking about one number. You're talking about a whole set of numbers, such as $4$4, $2$2, $3$3, $0$0, $-1$1, $-1000$1000, $\frac{1}{2}$12. All of these numbers are less than or equal to $4$4.

So should we plot the inequality $x\le4$x4 like this?

The problem with this is that it's not just these numbers that are less than or equal to $4$4. It's the countless collection of numbers in between, along with every number down towards negative infinity. One endless line of numbers to the left of $4$4. We represent this as a ray from $4$4 pointing all the way left across the entire number line.

$x\le4$x4

Now, what if instead we wanted to plot the inequality $x<4$x<4. The only difference now is that $x$x cannot be equal to $4$4. To represent this, we plot the ray in the same way, but since $4$4 is no longer included, we have a hole where $4$4 is supposed to be.

$x<4$x<4

For $x\ge4$x4 or $x>4$x>4, we just flip the direction of the ray!

$x\ge4$x4
$x>4$x>4

 

Practice questions

Question 4

Consider the inequality $\frac{-8-3x}{2}\le5$83x25.

  1. Solve the inequality.

  2. Hence, plot the inequality $\frac{-8-3x}{2}\le5$83x25 on the number line below.

    -10-50510

Question 5

Consider the inequality $-4\left(5x-3\right)>52$4(5x3)>52.

  1. Solve the inequality.

  2. Hence, plot the inequality $-4\left(5x-3\right)>52$4(5x3)>52 on the number line below.

    -10-50510

Question 6

Consider the inequality: $\frac{3x}{8}+9>21$3x8+9>21

  1. Solve the inequality.

  2. Hence, plot the inequality $\frac{3x}{8}+9>21$3x8+9>21 on the number line below.

    25303540

Outcomes

8.18

Solve multistep linear inequalities in one variable with the variable on one or both sides of the inequality symbol, including practical problems, and graph the solution on a number line

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