4. Equations & Inequalities

Lesson

An equation is a mathematical sentence stating that two expressions are equal. That is, they have the same value. We can think of it as being balanced.

Here are some examples:

$3x+2=4$3`x`+2=4

$4.25=3\frac{1}{2}+n$4.25=312+`n`

$\frac{y}{3}=2y+5$`y`3=2`y`+5

Equations often contain letters or symbols are used to represent an unknown quantity. These symbols are called variables. As seen above, variables can appear on one or both sides of an equation.

An equation is like a sentence. We would like to translate a written sentence to a mathematical sentence. Where we see "is" or "equals", we will put an equals sign.

Adam thought of a number, doubled it and added $5$5 to get $13$13. Write an equation that represents this scenario.

**Think: **There is an unknown quantity, "a number", so we should use a variable to represent it. Let's use $n$`n`.

**Do: **

$n\times2+5$n×2+5 |
$=$= | $13$13 |
We take a number, double it and then add $5$5, this is equal to $13$13 |

$2n+5$2n+5 |
$=$= | $13$13 |
We prefer to write a product of a number and variable with the number first |

What scenario could the equation $3a=2b+1$3`a`=2`b`+1 be representing?

**Think:** There is certainly not a single correct answer here, so here is one possible scenario.

We should start by stating what $a$`a` and $b$`b` represent.

**Do: **Let $a$`a` be the number of apples and $b$`b` be the number of bananas. The cost of $3$3 apples is $1$1 dollar more than the cost of $2$2 bananas.

**Reflect: **How many different scenarios can you come up with? What do all the scenarios have in common? What is different?

Is $5r-15=0$5`r`−15=0 an expression or an equation?

It is an expression.

AIt is an equation.

BIt is an expression.

AIt is an equation.

B

A number (call it $n$`n`) plus four equals seven.

Write the sentence using mathematical symbols.

What is the value of $n$

`n`?

Roxanne has been out picking flowers, and has $40$40 in total. When she returns, she puts them in $5$5 different vases.

If she puts $p$`p` flowers in each vase, rewrite the following sentence using algebra:

"There are $5$5 groups of $p$

`p`flowers, which make $40$40 in total."

We say that a value for a variable is a solution to an equation if we can substitute it into the equation and it makes the number sentence true.

For example, if we wanted to find the solution to the equation $x+1=3$`x`+1=3, we want to find a value for $x$`x` that makes that equation true. This statement is true when $x=2$`x`=2 because $2+1=3$2+1=3.

When we're figuring out whether a value is a solution, we need to see whether the left-hand side of the equation is the same as the right-hand side. We can think of it as a see-saw.

For example, if we have the equation $x+12=20$`x`+12=20, we could think of it visually as:

If $12$12 was removed from the left-hand side of the seesaw, it would look unbalanced like this:

So how do we balance the equation again? We need to remove $12$12 from the right-hand side as well:

So $x=8$`x`=8 satisfies the equation $x+12=20$`x`+12=20 because $8+12=20$8+12=20.

Determine if $b=47$`b`=47 is a solution of $b+48=96$`b`+48=96.

**a)** Find the value of the left-hand side of the equation when $b=47$`b`=47.

**Think:** We need to substitute $47$47 into the equation for $b$`b`.

**Do:**

$b+48$b+48 |
$=$= | $b+48$b+48 |

$=$= | $47+48$47+48 | |

$=$= | $95$95 |

The left-hand side is $95$95.

**b)** Is $b=47$`b`=47 a solution of $b+48=96$`b`+48=96?

**Think:** Is the left-hand side equal to the right-hand side in this equation?

**Do:** No, $b=47$`b`=47 is not a solution of $b+48=96$`b`+48=96.

There are two rectangular-shaped pools at the local aquatic center. Each pool has a length that is triple its width. Pool 1 has a perimeter of $256$256 meters.

Let the width of the pools be represented by $w$

`w`. Which of the following equations represents the perimeter of each pool in terms of $w$`w`?$2w+3w=256$2

`w`+3`w`=256A$3w+3w+3w=256$3

`w`+3`w`+3`w`=256B$w+3w=256$

`w`+3`w`=256C$w+w+3w+3w=256$

`w`+`w`+3`w`+3`w`=256D$2w+3w=256$2

`w`+3`w`=256A$3w+3w+3w=256$3

`w`+3`w`+3`w`=256B$w+3w=256$

`w`+3`w`=256C$w+w+3w+3w=256$

`w`+`w`+3`w`+3`w`=256DThe width of Pool 2 is $32$32 meters. Find its perimeter.

Is the width of Pool 1 also $32$32 meters?

no

Ayes

Bno

Ayes

B

We want to determine if $b=8$`b`=8 is the solution of $8b=63$8`b`=63.

Find the value of the left-hand side of the equation when $b=8$

`b`=8.Is $b=8$

`b`=8 the solution of $8b=63$8`b`=63?yes

Ano

Byes

Ano

B