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4.09 Problem solving with inequalities


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.

  • $>$>- 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?





    There is no largest value.








    There is no largest value.




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.



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.


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. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

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