This problem asks you to prove that the set S is exactly the set of strings over {a, b} which do not contain two or more consecutive a's. In other words, you will prove that x ∈ S if and only if x does not contain two consecutive a's. The two directions of the "if and only if" are proven separately. (a) Use structural induction to prove that if a string x ∈ S, then x does not have two or more consecutive a's

Solve the following inequalities and show the solution on a number line. 2x+3< 3(4x+5)

Paula makes stained-glass windows and sells them to boutique stores. if her costs total $12,000 per year plus $4 per window for the frame. how many windows must she produce to earn a profit of at least $48,000 in one year if she sells the windows for $28 each? 1. define a variable for the situation. 2. write an inequality that represents her profit. note: revenue is money coming in. cost is money going out. profit is the difference between the revenue and the cost. in other words: revenue - costs profit 3.using words, describe how many windows she must sell to have a profit of at least $48,000.

3c2 + 2d)(–5c2 + d) select all of the partial products for the multiplication problem above.