Stewart Precalc 6e Chapter 1 Test

(a) Graph the intervals \((-5, 3]\) and \((2, \infty)\) on the real number line. (b) Express the inequalities \(x \leq 3\) and \(-1 \leq x < 4\) in interval notation. (c) Find the distance between \(-7\) and \(9\) on the real number line.

Evaluate each expression. (a) \((-3)^4\) (b) \(-3^4\) (c) \(3^{-4}\) (d) \(\dfrac{5^{23}}{5^{21}}\) (e) \(\left(\dfrac{2}{3}\right)^{-2}\) (f) \(16^{-\dfrac{3}{4}}\)

Write each number in scientific notation. (a) \(186{,}000{,}000{,}000\) (b) \(0.0000003965\)

Simplify each expression. Write your final answer without negative exponents. (a) \(\sqrt{200} - \sqrt{32}\) (b) \((3 a^3 b^3)(4 a b^2)^2\) (c) \(\left(\dfrac{3 x^{\dfrac{3}{2}} y^3}{x^2 y^{-\dfrac{1}{2}}}\right)^{-2}\) (d) \(\dfrac{x^2 + 3x + 2}{x^2 - x - 2}\) (e) \(\dfrac{x^2}{x^2 - 4} - \dfrac{x + 1}{x + 2}\) (f) \(\dfrac{\dfrac{y}{x} - \dfrac{x}{y}}{\dfrac{1}{y} - \dfrac{1}{x}}\)

Rationalize the denominator and simplify: \(\dfrac{\sqrt{10}}{\sqrt{5}}\)

Perform the indicated operations and simplify. (a) \(3(x + 6) + 4(2x - 5)\) (b) \((x + 3)(4x - 5)\) (c) \((\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})\) (d) \((2x + 3)^2\) (e) \((x + 2)^3\)

Factor each expression completely. (a) \(4 x^2 - 25\) (b) \(2 x^2 + 5 x - 12\) (c) \(x^3 - 3 x^2 - 4 x + 12\) (d) \(x^4 + 27 x\) (e) \(3 x^{\dfrac{3}{2}} - 9 x^{\dfrac{1}{2}} + 6 x^{-\dfrac{1}{2}}\) (f) \(x^3 y - 4 x y\)

Find all real solutions. (a) \(x + 5 = 14 - \dfrac{1}{2} x\) (b) \(\dfrac{2 x}{x + 1} = \dfrac{2 x - 1}{x}\) (c) \(x^2 - x - 12 = 0\) (d) \(2 x^2 + 4 x + 1 = 0\) (e) \(\sqrt{3 - \sqrt{x + 5}} = 2\) (f) \(x^4 - 3 x^2 + 2 = 0\) (g) \(3 |x - 4| = 10\)

Mary drove from Amity to Belleville at a speed of 50 mi/h. On the way back, she drove at 60 mi/h. The total trip took \(4 \dfrac{2}{5}\) h of driving time. Find the distance between these two cities.

A rectangular parcel of land is 70 ft longer than it is wide. Each diagonal between opposite corners is 130 ft. What are the dimensions of the parcel?

Solve each inequality. Write the answer using interval notation, and sketch the solution on the real number line. (a) \(-4 < 5 - 3 x \leq 17\) (b) \(x(x - 1)(x + 2) > 0\) (c) \(|x - 4| < 3\) (d) \(\dfrac{2 x - 3}{x + 1} \leq 1\)

A bottle of medicine is to be stored at a temperature between \(5^{\circ} C\) and \(10^{\circ} C\). What range does this correspond to on the Fahrenheit scale? Note: Fahrenheit (\(F\)) and Celsius (\(C\)) temperatures satisfy the relation \(C = \dfrac{5}{9}(F - 32)\).

For what values of \(x\) is the expression \(\sqrt{6 x - x^2}\) defined as a real number?

Solve the equation and the inequality graphically. (a) \(x^3 - 9 x - 1 = 0\) (b) \(x^2 - 1 \leq |x + 1|\)

(a) Plot the points \(P(0, 3)\), \(Q(3, 0)\), and \(R(6, 3)\) in the coordinate plane. Where must the point \(S\) be located so that \(P Q R S\) is a square? (b) Find the area of \(P Q R S\).

(a) Sketch the graph of \(y = x^2 - 4\). (b) Find the \(x\)- and \(y\)-intercepts of the graph. (c) Is the graph symmetric about the \(x\)-axis, the \(y\)-axis, or the origin?

Let \(P\) and \(Q\) be two points (specific coordinates were not captured in OCR). (a) Plot \(P\) and \(Q\) in the coordinate plane. (b) Find the distance between \(P\) and \(Q\). (c) Find the midpoint of the segment \(P Q\). (d) Find the slope of the line that contains \(P\) and \(Q\). (e) Find the perpendicular bisector of the line that contains \(P\) and \(Q\). (f) Find an equation for the circle for which the segment \(P Q\) is a diameter.

Find the center and radius of each circle and sketch its graph. (a) \(x^2 + y^2 = 25\) (b) \((x - 2)^2 + (y + 1)^2 = 9\) (c) \(x^2 + 6 x + y^2 - 2 y + 6 = 0\)

Write the linear equation \(2 x - 3 y = 15\) in slope-intercept form, and sketch its graph. What are the slope and \(y\)-intercept?

Find an equation for the line with the given property. (a) It passes through the point \((3, -6)\) and is parallel to the line \(3 x + y - 10 = 0\). (b) It has \(x\)-intercept 6 and \(y\)-intercept 4.

A geologist uses a probe to measure the temperature \(T\) (in \(^{\circ} C\)) of the soil at various depths below the surface, and finds that at a depth of \(x\) cm, the temperature is given by the linear equation \(T = 0.08 x - 4\). (a) What is the temperature at a depth of one meter (100 cm)? (b) Sketch a graph of the linear equation. (c) What do the slope, the \(x\)-intercept, and \(T\)-intercept of the graph of this equation represent?

The maximum weight \(M\) that can be supported by a beam is jointly proportional to its width \(w\) and the square of its height \(h\), and inversely proportional to its length \(L\). (a) Write an equation that expresses this proportionality. (b) Determine the constant of proportionality if a beam 4 in. wide, 6 in. high, and 12 ft long can support a weight of 4800 lb. (c) If a 10-ft beam made of the same material is 3 in. wide and 10 in. high, what is the maximum weight it can support?