Inequalities

Ontario 10 Academic (MPM2D)

Properties of inequality

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