91Ó°ÊÓ

Factor the expression by grouping terms. $$x^{3}+x^{2}+x+1$$

Find an equation of the circle that satisfies the given conditions. Center \((2,-1) ;\) radius 3

Find all real solutions of the equation. $$\frac{1}{x-1}+\frac{1}{x+2}=\frac{5}{4}$$

The manager of a weekend flea market knows from past experience that if she charges \(x\) dollars for a rental space at the flea market, then the number \(y\) of spaces she can rent is given by the equation \(y=200-4 x\) (a) Sketch a graph of this linear equation. (Remember that the rental charge per space and the number of spaces rented must both be nonnegative quantities.) (b) What do the slope, the \(y\) -intercept, and the \(x\) -intercept of the graph represent?

Absolute Value Inequalities Solve the absolute value inequality. Express the answer using interval notation and graph the solution set. $$8-|2 x-1| \geq 6$$

Rationalize the denominator. $$\frac{y}{\sqrt{3}+\sqrt{y}}$$

Is \(\frac{1}{2}+\sqrt{2}\) rational or irrational? Is \(\frac{1}{2} \cdot \sqrt{2}\) rational or irrational? Experiment with sums and products of other rational and irrational numbers. Prove the following. (a) The sum of a rational number \(r\) and an irrational number \(t\) is irrational. (b) The product of a rational number \(r\) and an irrational number \(t\) is irrational. [Hint: For part (a), suppose that \(r+t\) is a rational number \(q\), that is, \(r+t=q .\) Show that this leads to a contradiction. Use similar reasoning for part (b).]

Find all real solutions of the equation. $$\frac{x+5}{x-2}=\frac{5}{x+2}+\frac{28}{x^{2}-4}$$

Rationalize the denominator. $$\frac{2(x-y)}{\sqrt{x}-\sqrt{y}}$$

A small-appliance manufacturer finds that if he produces \(x\) toaster ovens in a month, his production cost is given by the equation $$y=6 x+3000$$ (where \(y\) is measured in dollars). (a) Sketch a graph of this linear equation. (b) What do the slope and \(y\) -intercept of the graph represent?