Chapter 1.7, Problem 1E

(a) Let a be a negative real number. Prove that if a is a solution to the equation $x^{\text{2}}-x-\text{6}=0$ , then a is a solution to $x^{\text{3}}+\text{2}{x}^{\text{2}}+x+\text{2}=0$ .
(b) Let x be a real number. Prove that $0\le x\le \text{3}$ implies $x+\text{1}\ge {(x-\text{1})}^{\text{2}}$ .
(c) Let a and b be positive integers such that $a\ne 0$ . Prove that if a does not divide b, then there is no positive integer x that is a solution to $a{x}^{\text{2}}+bx+b-a=0$ .
(d) Prove that if x is a real number and $x\ge \text{1}$ , then $\frac{3\left|x-2\right|}{x}\le 4$ .
(e) Prove that no point inside the circle ${(x-\text{ 3})}^{\text{2}}+y^{\text{2}}=\text{6}$ is on the line $y=x+\text{1}$ .
(f) Let x be a real number. Prove that if $x^{\text{2}}+\text{7}x=\text{9}x+\text{15}$ , then $x\le \text{2}$ or $\frac{x-4}{x-3}>0$ .
(g) Suppose that x is a real number. Prove that $-x^{\text{2}}+\text{7}x-\text{1}0<0$ if and only if $x<\text{2}$ or $x>\text{5}$ .
(h) Prove that if two nonvertical lines are perpendicular, then the product of their slopes is -1. (Recall that nonvertical lines are those lines in the plane that have slope.)
(i) Prove that there exists a point $(x,y)$ inside the circle ${(x-\text{3})}^{\text{2}}+{(y-2)}^{\text{2}}=\text{13}$ such that $x<0$ and $y>\text{3}.\text{99}$ .
(j) Prove that there is a pair, and only one pair, $(x,y)$ of real numbers such that $0\le x\le \text{2},0\le y\le \text{1}$ , and $\text{3}{x}^{\text{2}}+\text{2}{y}^{\text{2}}\ge \text{14}$ .