Chapter 1: Fundamentals

We have learned that addition and multiplication are both commutative operations.

(a) Is subtraction commutative?

(b) Is division of nonzero real numbers commutative?

(c) Are the actions of putting on your socks and putting on your shoes commutative?

(d) Are the actions of putting on your hat and putting on your coat commutative? (e) Are the actions of washing laundry and drying it commutative?

Factor the expression completely. Begin by factoring out the lowest power of each common factor.

${\left(x-1\right)}^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\left(x-1\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Other Equations Find all real solutions of the equation.

$\sqrt{8x-1}=3$

Find an equation of the circle that satisfies the given conditions.

Endpoints of a diameter are$P(-1,3)\mathrm{and}Q(7,-5)$

Rationalize the numerator.

$\frac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x}\sqrt{x+h}}$

A phrase describing a set of real numbers is given. Express the phase as an inequality involving an absolute value.

All real numbers x at most 4 units from 2

At the surface of the ocean the water pressure is the same as the air pressure above the water, $15\mathrm{lb}/{\mathrm{in}}^2$. Below the surface the water pressure increases by $4.34\mathrm{lb}/{\mathrm{in}}^2$for every 10 ft of descent.

(a)Find an equation for the relationship between pressure and depth below the ocean surface.

(b)Sketch a graph of this linear equation.

(c)What do the slope and y-intercept of the graph represent?

(d)At what depth is the pressure$100\mathrm{lb}/{\mathrm{in}}^2$?

Find all real solutions of the equation.

$\sqrt{2x-1}=\sqrt{3x-5}$

A set of real numbers is graphed. Find an inequality involving an absolute value that describes the set.

Suppose that the graph of the outdoor temperature over a certain period of time is a line. How is the weather changing if the slope of the line is positive? If it is negative? If it is zero?