We have to give one last note on interval notation before moving on to solving inequalities. Always recall that while we are writing down an interval notation for inequality that the number onto the left has to be the smaller of the two.
Now it's time to begin thinking about solving linear inequalities. We will employ the following set of facts in our solving of inequalities. Note down that the facts are given for <. However we can write down an equivalent set of facts for the remaining three inequalities.
1. If a < b then a + c < b + c and a - c < b - c for any number c. In other term, we can add or subtract a number to both of sides of the inequality & we don't vary the inequality itself.
2. If a < b and c > 0 then ac
3. If a < b and c<0 then ac > bc and a/c > b/c . In this case, unlike the earlier fact, if c is negative we have to flip the direction of the inequality while we multiply or divide both sides by the inequality through c.
These are closely the similar facts that we utilized to solve linear equations. The single real exception is the third fact. It is the important issue as it is frequently the most misused and/or forgotten fact in solving inequalities.
If you aren't certain that you believe that the sign of c matters for the second & third fact assume the following number instance.
-3 < 5
This is a true inequality. Now multiply both of sides by 2 and by -2.
- 3 < 5 - 3 < 5
-3( 2) < 5 ( 2) -3 ( -2) < 5 ( -2)
- 6 < 10 6 < -10
Sure enough, while multiplying by a +ve number the direction of the inequality remains the similar, however while multiplying by a -ve number the direction of the inequality does change.