Ontario 10 Academic (MPM2D)
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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+77 $=$= $12-7$127 (subtracting $7$7 from both sides)
$x$x $=$= $5$5 (simplifying both sides)

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

Exploration

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$93<153.

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.

Practice questions

Question 1

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

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

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

    Yes

    A

    No

    B

    Yes

    A

    No

    B

question 2

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

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

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

    Yes

    A

    No

    B

    Yes

    A

    No

    B

question 3

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

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

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

    Yes

    A

    No

    B

    Yes

    A

    No

    B

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