14
Adding and multiplying inequalities
Textbook: pp. 75–79
GoalKnow how to add and multiply inequalities term by term and use this to estimate values of expressions.
New words
term-by-term addition · hadlab qo‘shishterm-by-term multiplication · hadlab ko‘paytirishestimating · baholashdouble inequality · qo‘sh tengsizlik
Explanation
Inequalities of the same sign can be added term by term: if a > b and c > d then a + c > b + d. Proof: (a + c) - (b + d) = (a - b) + (c - d) is a sum of positive numbers. Inequalities of the same sign whose sides are all positive can also be multiplied term by term: if a > b > 0 and c > d > 0 then ac > bd. With numbers that are not positive this rule can fail. With these rules we estimate sums, products, perimeters and areas. To estimate a difference, first multiply an inequality by -1, then add.
Worked examples
If 2 < x < 3 and 4 < y < 5 then 6 < x + y < 8 and 8 < xy < 15 (all numbers positive). From -5 < -y < -4 we get -3 < x - y < -1.
If a rectangle has sides 4 < a < 5 and 2 < b < 3, then P = 2(a + b): 6 < a + b < 8, so 12 < P < 16; and the area S = ab satisfies 8 < S < 15.
Class activity
“The estimators”: each pair gives the sides of a box within bounds (for example 20 < a < 21) and another pair estimates the volume and surface.
Practice
1
If 2 < a < 3 and 1 < b < 2, estimate a + b.
3 < a + b < 5
2
Under the same conditions estimate ab.
2 < ab < 6
3
If 5 < x < 7, estimate 2x - 1.
9 < 2x - 1 < 13
4
Why can’t inequalities with negative numbers be multiplied term by term? Give an example.
Multiplying -3 < -2 and -5 < -1 gives 15 and 2, but 15 > 2: the sign is broken.