2.07 Linear inequalities

We have seen that it is easiest to plot an inequality on a number line by first solving the inequality. We have also looked at solving simple inequalities.

Suppose we want to plot the solutions to the inequality 2(3+x)\lt on a number line. That is, we want to plot the values of x which can be added to 3 and then doubled to result in a number less than 8.

To solve this inequality, we want to undo these operations in reverse order. That is, we can solve this inequality by first dividing both sides by 2, then subtracting 3 from both sides:

In this case, we arrive at the result x<1. We can test some values in the original inequality to see if this is the right solution set - let's say x=0 and x=2.

• When x=0, we have 2\left(3+x\right)=2\left(3+0\right)=6, which is less than 8.
• When x=2, we have 2\left(3+x\right)=2\left(3+2\right)=10, which is not less than 8.

So our result of x\lt 1 seems to be correct.

We can now plot the solutions on a number line as follows, using a hollow circle for the endpoint (since x=1 is not included in the solutions):

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, using the properties of inequality.

When plotting an inequality:

• The symbols < and > don't include the end point, which we show with a hollow circle.

• The symbols \geq and \leq do include the endpoint, which we show with a filled circle.

Examples

We have now looked at solving inequalities that involve one or two steps to solve. We're now going to take a look at how we can use inequalities to solve problems given a written description.

Much as with solving equations from written descriptions, there are certain keywords or phrases to look out for. When it comes to inequalities, we now have a few extra keywords and phrases to represent the different inequality symbols.

Here are the phrases and keywords:

• \gt - greater than, more than.

• \geq - greater than or equal to, at least, no less than.

• \lt - less than.

• \leq - less than or equal to, at most, no more than.

Examples

Example 3

Consider the following situation: "2 less than 4 groups of p is no more than 18".

a

Construct and solve the inequality described above.

Worked Solution

Create a strategy

Translate the phrases into mathematical symbols to set up the inequality. Solve the inequality by isolating x on side of the inequality.

Apply the idea

The phrase "no more than" means the same as "less than or equal to", "less than" means substracting from, and "groups of" means a multiplication operation.

\displaystyle 4p-2	\displaystyle \leq	\displaystyle 18	Write the inequality
\displaystyle 4p-2+2	\displaystyle \leq	\displaystyle 18+2	Subtract 2 from both sides
\displaystyle 4p	\displaystyle \leq	\displaystyle 20	Simplify
\displaystyle \dfrac{4p}{4}	\displaystyle \leq	\displaystyle \dfrac{20}{4}	Divide both sides by 4
\displaystyle p	\displaystyle \leq	\displaystyle 5	Simplify

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b

What is the largest value of p that satisfies this condition?

A

p=5

B

p=-5

C

There is no largest value.

D

p=4

Worked Solution

Create a strategy

Determine the largest value of the inequality from part (a) by recalling the definition of inequality symbols.

Apply the idea

Using a number line, we can plot p\leq 5 as:

The inequality p\leq5 means that p can take any value that is less than or equal to 5. So the largest value of p that can satisfy the inequality is 5. The correct answer is option A.

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Example 4

Lachlan is planning on going on vacation. He has saved \$2118.40, and spends \$488.30 on his airplane ticket.

a

Let x represent the amount of money Lachlan spends on the rest of his holiday. Write an inequality to represent the situation, and then solve for x.

Worked Solution

Create a strategy

The amount of money that he can spend on his holiday is up to but not more than the difference between his savings and the amount spent on the airplane ticket. Translate this information into mathematical symbols.

Apply the idea

The phrase "up to but not more than" means that we are going to use the \leq symbol and the phrase "difference between" means subtraction.

\displaystyle x	\displaystyle \leq	\displaystyle 2\,118.40 - 488.30	Write the inequality
\displaystyle x	\displaystyle \leq	\displaystyle \$1\,630.10	Evaluate

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b

What is the most that Lachlan could spend on the rest of his holiday?

Worked Solution

Create a strategy

Determine the largest value of the inequality from part (a) by recalling the definition of inequality symbol.

Apply the idea

The inequality x \leq\$1\,630.10 means that x can take any value that is less than or equal to \$1\,630.10. So the maximum amount that Lachlan can spend on his holiday is \$1\,630.10.

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Example 5

At a sport clubhouse the coach wants to rope off a rectangular area that is adjacent to the building. He uses the length of the building as one side of the area, which measures 26 meters. He has at most 42 meters of rope available to use.

If the width of the roped area is W, form an inequality and solve for the range of possible widths.

Worked Solution

Create a strategy

Translate the given information into mathematical symbols and use the perimeter equation to solve for the possible width of the rope.

Apply the idea

One side of the roped area is the length of the clubhouse. So the situation looks something like this:

The length of the roped area is equal to the length of the clubhouse which is 26 meters.

\displaystyle \text{width + width + length}	\displaystyle =	\displaystyle \text{Perimeter}	Rope perimeter is equal to the sum of two widths and the length
\displaystyle 2W+26	\displaystyle \leq	\displaystyle 42	Substitute the values and variables
\displaystyle 2W + 26 - 26	\displaystyle <	\displaystyle 42 - 26	Subtract 26 from both sides
\displaystyle 2W	\displaystyle \leq	\displaystyle 16	Simplify
\displaystyle \dfrac{2W}{2}	\displaystyle \leq	\displaystyle \dfrac{16}{2}	Divide both sides by 2
\displaystyle W	\displaystyle \leq	\displaystyle 8\text{ m}	Simplify

The maximum width should be 8\text{ m} to be able to use all of the 42 \text{ m} rope and we will have enough.

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So far we have looked at how to solve inequalities for all of the possible solutions, and we found that the process was very similar to how we solve equations. We don't always need to solve for all of the possible solutions, however. Sometimes we just want to know if a particular value will satisfy an inequality or not.

Think about the inequality given by p-7< 12. There is an interval of values that p can take to make the inequality true.

For example, we can test whether p=20 satisfies p-7<12 by simply substituting that value into it and see what happens.

So, if p=20, the expression becomes 20-7<12 or, in other words, 13<12 which is clearly not true. This means that p=20 is not in the solution set. It doesn't satisfy the inequality.

On the other hand, a number like p=10 is in the solution set because 10-7<12.

Examples