# 4.09 Problem solving with inequalities

Lesson

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+17.

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

 $2x+1$2x+1 $\ge$≥ $7$7 Inequality we set up $2x$2x $\ge$≥ $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".

1. Construct and solve the inequality described above.

2. 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

$p=5$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.

1. 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.

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

##### Question 4

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

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

### Outcomes

#### 7eE.B.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.

#### 7eE.B.4b

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.