3. Equations & Inequalities

Lesson

When we manipulate with equalities, we can apply the same operation to both sides and the equality statement remains true. Take the following equality:

$x+7$x+7 |
$=$= | $12$12 |

We can subtract both sides of the equation in order to find the value of $x$`x`. This is because both sides of the equation are identical, so what we do to one side, we should do to the other side.

$x+7$x+7 |
$=$= | $12$12 | (rewriting the equation) |

$x+7-7$x+7−7 |
$=$= | $12-7$12−7 | (subtracting $7$7 from both sides) |

$x$x |
$=$= | $5$5 | (simplifying both sides) |

When working with inequalities, this is not necessarily always the case.

Consider the inequality $9<15$9<15. If we add or subtract both sides by any number, say $3$3, we can see that the resulting inequality remains true. More specifically we can write $9+3<15+3$9+3<15+3 and $9-3<15-3$9−3<15−3.

Adding $3$3 to $9$9 and $15$15. |

Subtracting $3$3 from $9$9 and $15$15. |

Now consider if we multiply or divide both sides of the inequality by $3$3. We get $9\times3<15\times3$9×3<15×3 and $\frac{9}{3}<\frac{15}{3}$93<153. These statements are true, since we increase (or decrease) $9$9 and $15$15 by the same positive factor, so the signs of each side are unchanged.

However, if we had chosen a negative number, like $-3$−3, the signs of each side are changed and we must swap the inequality sign around. So the correct statements are $9\times\left(-3\right)>15\times\left(-3\right)$9×(−3)>15×(−3) and $-\frac{9}{3}>-\frac{15}{3}$−93>−153.

Consider the following statement: $7<10$7<10

Add $6$6 to both sides of the inequality and simplify.

After adding $6$6 to both sides, does the inequality still hold true?

Yes

ANo

BYes

ANo

B

Consider the following statement: $5<7$5<7

Multiply both sides of the inequality by $2$2 and simplify.

After multiplying both sides by $2$2, does the inequality still hold true?

Yes

ANo

BYes

ANo

B

Consider the following statement: $6<10$6<10

Multiply both sides of the inequality by $-4$−4 and simplify. Do not change the sign of the inequality.

After multiplying both sides by $-4$−4, does the inequality still hold true?

Yes

ANo

BYes

ANo

B