4. Equations & 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.

**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$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 groups 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 |
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.

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 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.

Use variables to represent quantities in a real-world 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, px + q ≥ r, px + q ≤ r, or px + q < r, where p, q, and r are rational numbers. Graph the solution set of the inequality on the number line and interpret it in the context of the problem.