Inequalities

NZ Level 6 (NZC) Level 1 (NCEA)

Problem solving with inequalities

Lesson

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.

Phrases

- $>$>- 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.

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$2`x`, and the sum of this and $1$1 is $2x+1$2`x`+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$2`x`+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.

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`=5A$p=-5$

`p`=−5BThere is no largest value.

C$p=4$

`p`=4D$p=5$

`p`=5A$p=-5$

`p`=−5BThere is no largest value.

C$p=4$

`p`=4D

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?

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 metres. He has at most $42$42 metres 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.

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

Apply algebraic procedures in solving problems