Properties of inequality
When we manipulate with equalities, we can apply the same operation to both sides and the equality statement remains true. Take the following equality:
| $$x+7 | $$= | $$12 |
We can subtract both sides of the equation in order to find the value of $$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.
| $$x+7 | $$= | $$12 | (rewriting the equation) |
| $$x+7−7 | $$= | $$12−7 | (subtracting $$7 from both sides) |
| $$x | $$= | $$5 | (simplifying both sides) |
When working with inequalities, this is not necessarily always the case.
Exploration
Consider the inequality $$9<15. If we add or subtract both sides by any number, say $$3, we can see that the resulting inequality remains true. More specifically we can write $$9+3<15+3 and $$9−3<15−3.
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| Adding $$3 to $$9 and $$15. |
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| Subtracting $$3 from $$9 and $$15. |
Now consider if we multiply or divide both sides of the inequality by $$3. We get $$9×3<15×3 and $$93<153. These statements are true, since we increase (or decrease) $$9 and $$15 by the same positive factor, so the signs of each side are unchanged.
However, if we had chosen a negative number, like $$−3, the signs of each side are changed and we must swap the inequality sign around. So the correct statements are $$9×(−3)>15×(−3) and $$−93>−153.
Practice questions
Question 1
Consider the following statement: $$7<10
Add $$6 to both sides of the inequality and simplify.
After adding $$6 to both sides, does the inequality still hold true?
Yes
ANo
B
question 2
Consider the following statement: $$5<7
Multiply both sides of the inequality by $$2 and simplify.
After multiplying both sides by $$2, does the inequality still hold true?
Yes
ANo
B
question 3
Consider the following statement: $$6<10
Multiply both sides of the inequality by $$−4 and simplify. Do not change the sign of the inequality.
After multiplying both sides by $$−4, does the inequality still hold true?
Yes
ANo
B