Algebra Answers

Prove that if A and B are any two sets such that A ⊆ B, then A ∪ B = B
i) by direct method;
ii) by proving its contrapositive;
iii) by contradiction.

Solve the equation x^3+ 2x^2- 2x -1 = 0. using Cardano’s method.

If the sum of the roots of a cubic equation is 3, the sum of the squares of the roots is 11 and the
sum of the cubes of the roots is 27, find the equation. Also, solve the equation and find all its
roots.

Express the following problem situation as a linear system, and then solve it by substitution. Also
show the linear system geometrically.
A manufacturer produces two types of cupboards – deluxe and regular. Each deluxe cupboard
requires 12 worker hours to cut and assemble, and 5 worker hours to finish. Each regular
cupboard requires 8 worker hours to cut and assemble, and 3 hours to finish. On a daily basis,
the manufacturer has available 440 worker hours for cutting and assembling, and 175 worker
hours for finishing. How many cupboards of each type should be produced so that all the work
power is utilized?

Give the geometric representation of the Cartesian product {–2, 1} × [–2, 1].

Consider the sets. A = {3x │ x∈ Z}, B = {x ∈ Q │ –5 ≤ x ≤ 10} and C = {factors of 20 in Z}.
Represent A ∩ B, Bc ∪C and C \ A by the listing method and in a single Venn diagram.

Given a, b > 0 such that ab = 18, check whether their sum is greatest when a = b.

Let m, n, x, y, z be positive real numbers, with x + y + z = 1. Prove that
4 4 4
2
x y z 1
(mx ny) (my nx) (my nz) (mz ny) (mz nx) (mx nz) 3(m n)
+ + ≥
+ + + + + + +
[Hint: Apply the AM ≥ GM inequality to each term in the LHS, and then apply the CS
inequality.]

Write an equation in slope-intercept form for the line with slope -3and y-intercept -2 .

Dissmore's has reduced prices by 19% on everything to thank their customers for their patronage. If Ben & Jerry's ® Phish Food TM ice cream cost $2.99 before the price reduction, how much does it cost now? (Round your answer to the nearest penny.)
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