topic badge
New Zealand
Level 6 - NCEA Level 1

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+17.

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".

  1. Construct and solve the inequality described above.

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





    There is no largest value.




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.

  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 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.

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



Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns


Apply algebraic procedures in solving problems

What is Mathspace

About Mathspace