4. Equations & Inequalities

Lesson

We want to keep equations balanced so that the two sides of the equals sign remain equivalent. If we don't we could change what the equation means. Think of a balanced set of scales. The scale remains level when the weights on both side of the scales are even. The same thing happens with equations.

If we add a weight to one side and not to the other, then the scales will no longer be balanced.

This applet represents the equation $x=5$`x`=5. What equivalent equations can you make by doing the same thing to both sides? See if you can come up with $4$4 different equations.

Beginning with the equation $x=8$`x`=8, write the new equation produced by adding $2$2 to both sides.

**Think:** We will start with $x=8$`x`=8 and then to keep the equation balanced, we must do the same thing to both sides.

**Do:**

$x$x |
$=$= | $8$8 |
Given equation |

$x+2$x+2 |
$=$= | $8+2$8+2 |
Adding $2$2 to both sides |

$x+2$x+2 |
$=$= | $10$10 |
Simplify - perform the addition |

Beginning with the equation $x=-99$`x`=−99, write the new equation produced by dividing both sides by $11$11.

**Think:** We will start with $x=-99$`x`=−99 and then to keep the equation balanced, we must do the same thing to both sides.

**Do:**

$x$x |
$=$= | $-99$−99 |
Given equation |

$x\div11$x÷11 |
$=$= | $\left(-99\right)\div11$(−99)÷11 |
Dividing both sides by $11$11 |

$\frac{x}{11}$x11 |
$=$= | $-9$−9 |
Simplify - write as a fraction and perform the division |

Beginning with the equation $33x=99$33`x`=99, write the new equation produced by dividing both sides by $11$11.

**Think:** We will start with $33x=99$33`x`=99 and then to keep the equation balanced, we must do the same thing to both sides.

**Do:**

$33x$33x |
$=$= | $99$99 |
Given equation |

$33x\div11$33x÷11 |
$=$= | $99\div11$99÷11 |
Dividing both sides by $11$11 |

$3x$3x |
$=$= | $9$9 |
Simplify - perform the division |

Remember

Keep equations balanced by always performing the exact same operation to both sides of the equation.

Beginning with the equation $x=14$`x`=14, write the new equation produced by subtracting $7$7 from both sides.

Beginning with the equation $x=99$`x`=99, write the new equation produced by dividing both sides by $11$11.

We can often solve for an unknown value by setting up an equation and then solving for the unknown value, often represented with a variable. To solve for the unknown variable, we will use opposite operations to get it by itself.

Suppose a person holding a dog steps on a scale. The number that shows on the scale is $75$75 kg. The person weighs $70$70 kg. First set up an equation for this scenario using $x$`x` as the weight of the dog, then solve the equation.

**Think**: We know that $\text{Person }+\text{Dog }=75$Person +Dog =75, that $\text{Person }=70$Person =70, and that $\text{Dog }=x$Dog =`x`. Using this we can set up an equation.

**Do: **

$70+x$70+x |
$=$= | $75$75 |
Filling in what we know |

$x+70$x+70 |
$=$= | $75$75 |
We often write the variable first |

$x+70-70$x+70−70 |
$=$= | $75-70$75−70 |
Subtract $70$70 from both sides to get $x$ |

$x$x |
$=$= | $5$5 |
Simplify |

The dog weighs $5$5 kg.

**Reflect**: What other strategies could you use to find the weight of the dog? Does this answer make sense?

Summary

- When asked to solve, we want to get the variable by itself on one side of the equals sign
- To keep everything balanced, we must do the same operations to both sides
- We should use opposite operations to solve
- Addition and subtraction are opposite operations
- Multiplication and division are opposite operations

Solve: $x+6=15$`x`+6=15

Solve: $21=x+13$21=`x`+13

Solve: $\frac{x}{8}=6$`x`8=6