We have now looked at solving inequalities that involve one step 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 worded problems, 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.

Keywords

$>$>- greater than, more than.
$\ge$≥- greater than or equal to, at least, no less than.
$<$<- less than.
$\le$≤- less than or equal to, at most, no more than.

Worked example

Question 1

Construct and solve an inequality for the following situation:

"The sum of $2$2 groups of $x$x and $1$1 is at least $7$7."

Think: "At least" means the same as "greater than or equal to". Also "lots of" means there is a multiplication, and "sum" means there is an addition.

Do: $2$2 groups of $x$x is $2x$2x, and the sum of this and $1$1 is $2x+1$2x+1. So altogether we have that "the sum of $2$2 groups of $x$x and $1$1 is at least $7$7" can be written as $2x+1\ge7$2x+1≥7.

We can now solve the inequality for $x$x:

$2x+1$2`x`+1	$\ge$≥	$7$7	Inequality we set up
$2x$2`x`	$\ge$≥	$6$6	Subtract $1$1 from both sides
$x$`x`	$\ge$≥	$3$3	Divide both sides by $2$2

So the possible values of $x$x are those that are greater than or equal to $3$3.

Practice questions

Question 2

Consider the following situation:

"$2$2 less than $4$4 groups of $p$p is no more than $18$18".

Construct and solve the inequality described above.
What is the largest value of $p$p that satisfies this condition?
$p=5$p=5
A
$p=-5$p=−5
B
There is no largest value.
C
$p=4$p=4
D

Question 3

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

Let $x$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 $x$x.
What is the most that $Lachlan$ could spend on the rest of his holiday?

Question 4

At a sports 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$26 meters. He has at most $42$42 meters of rope available to use.

A rectangle labeled Clubhouse is positioned above a larger dashed red rectangle labeled Rope. The Clubhouse rectangle is aligned at the top of the

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

Outcomes

7.EE.4

Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

7.EE.4.b

Solve word problems leading to inequalities of the form px q > r or px q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

4.09 Problem solving with inequalities