Problem solving with inequalities
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 key words or phrases to look out for. When it comes to inequalities, we now have a few extra key words and phrases to represent the different inequality symbols.
- $>$>- 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 lots 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 lots 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 lots 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$2x+1 | $\ge$≥ | $7$7 |
| $2x$2x | $\ge$≥ | $6$6 |
| $x$x | $\ge$≥ | $3$3 |
So the possible values of $x$x are those that are greater than or equal to $3$3.
Practice questions
Question 1
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
BThere is no largest value.
C$p=4$p=4
D
Question 2
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 3
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.
If the width of the roped area is $W$W, form an inequality and solve for the range of possible widths.