Stewart Precalc 6e Section 1.Review

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Stewart Precalc 6e Section 1.Review 0/139
1 Properties of Real Numbers · Level 1
State the property of real numbers being used: \(3 x + 2 y = 2 y + 3 x\).
2 Properties of Real Numbers · Level 1
State the property of real numbers being used: \((a + b)(a - b) = (a - b)(a + b)\).
3 Properties of Real Numbers · Level 1
State the property of real numbers being used: \(4(a + b) = 4 a + 4 b\).
4 Properties of Real Numbers · Level 1
State the property of real numbers being used: \((A + 1)(x + y) = (A + 1) x + (A + 1) y\).
5 Intervals - Inequality and Graph · Level 1
Express the interval in terms of inequalities, and then graph the interval: \([-2, 6)\).
6 Intervals - Inequality and Graph · Level 1
Express the interval in terms of inequalities, and then graph the interval: \((-\infty, 4]\).
7 Intervals - Inequality to Interval · Level 1
Express the inequality in interval notation, and then graph the corresponding interval: \(x \geq 5\).
8 Intervals - Inequality to Interval · Level 1
Express the inequality in interval notation, and then graph the corresponding interval: \(-1 < x < 5\).
9 Evaluating Expressions · Level 2
Evaluate the expression: \(|3 - abs(-9)|\).
10 Evaluating Expressions · Level 2
Evaluate the expression: \(1 - |1 - abs(-1)|\).
11 Evaluating Expressions - Exponents · Level 2
Evaluate the expression: \(2^{-3} - 3^{-2}\).
12 Evaluating Expressions - Roots · Level 1
Evaluate the expression: \(\sqrt[3]{-125}\).
13 Evaluating Expressions - Rational Exponents · Level 2
Evaluate the expression: \(216^{-\dfrac{1}{3}}\).
14 Evaluating Expressions - Rational Exponents · Level 2
Evaluate the expression: \(64^{\dfrac{2}{3}}\).
15 Evaluating Expressions - Radicals · Level 2
Evaluate the expression: \(\dfrac{\sqrt{242}}{\sqrt{2}}\).
16 Evaluating Expressions - Radicals · Level 2
Evaluate the expression: \(\sqrt[4]{4} \cdot \sqrt[4]{324}\).
17 Evaluating Expressions - Rational Exponents · Level 2
Evaluate the expression: \(2^{\dfrac{1}{2}} \cdot 8^{\dfrac{1}{2}}\).
18 Evaluating Expressions - Radicals · Level 1
Evaluate the expression: \(\sqrt{2} \cdot \sqrt{50}\).
19 Simplifying Expressions · Level 2
Simplify the expression: \(\dfrac{x^2 (2 x)^4}{x^3}\).
20 Simplifying Expressions · Level 3
Simplify the expression: \((a^2)^{-3} (a^3 b)^2 (b^3)^4\).
21 Simplifying Expressions · Level 3
Simplify the expression: \((3 x y^2)^3 \left(\dfrac{2}{3} x^{-1} y\right)^2\).
22 Simplifying Expressions - Rational Exponents · Level 3
Simplify the expression: \(\left(\dfrac{r^2 s^{\dfrac{4}{3}}}{r^{\dfrac{1}{3}} s}\right)^6\).
23 Simplifying Expressions - Radicals · Level 2
Simplify the expression: \(\sqrt[3]{(x^3 y)^2 y^4}\).
24 Simplifying Expressions - Radicals · Level 2
Simplify the expression: \(\sqrt{r^2 v^4}\).
25 Simplifying Expressions - Rational Exponents · Level 3
Simplify the expression: \(\left(\dfrac{9 x^3 y}{x^{-3}}\right)^{\dfrac{1}{2}}\).
26 Simplifying Expressions - Rational Exponents · Level 4
Simplify the expression: \(\left(\dfrac{x^{-2} y^3}{x^2 y}\right)^{-\dfrac{1}{2}} \left(\dfrac{x^3 y}{y^{\dfrac{1}{2}}}\right)^2\).
27 Simplifying Expressions - Rational Exponents · Level 3
Simplify the expression: \(\dfrac{8 r^{\dfrac{1}{2}} s^{-3}}{2 r^{-2} s^4}\).
28 Simplifying Expressions · Level 3
Simplify the expression: \(\dfrac{a b^2 c^{-3}}{2 a^3 b^{-4}}\).
29 Scientific Notation · Level 1
Write the number \(78{,}250{,}000{,}000\) in scientific notation.
30 Scientific Notation · Level 1
Write the number \(2.08 \times 10^{-8}\) in ordinary decimal notation.
31 Scientific Notation - Calculator · Level 2
If \(a \approx 0.00000293\), \(b \approx 1.582 \times 10^{-14}\), and \(c \approx 2.8064 \times 10^{12}\), use a calculator to approximate the number \(a \dfrac{b}{c}\).
32 Scientific Notation - Application · Level 2
If your heart beats \(80\) times per minute and you live to be \(90\) years old, estimate the number of times your heart beats during your lifetime. State your answer in scientific notation.
33 Factoring · Level 2
Factor the expression completely: \(12 x^2 y^4 - 3 x y^5 + 9 x^3 y^2\).
34 Factoring - Quadratic Trinomial · Level 2
Factor the expression completely: \(x^2 - 9 x + 18\).
35 Factoring - Quadratic Trinomial · Level 2
Factor the expression completely: \(x^2 + 3 x - 10\).
36 Factoring - Quadratic Trinomial · Level 2
Factor the expression completely: \(6 x^2 + x - 12\).
37 Factoring - Quadratic Trinomial · Level 2
Factor the expression completely: \(4 t^2 - 13 t - 12\).
38 Factoring - Perfect Square / Difference of Squares · Level 2
Factor the expression completely: \(x^4 - 2 x^2 + 1\).
39 Factoring - Difference of Squares · Level 1
Factor the expression completely: \(25 - 16 t^2\).
40 Factoring - Combined · Level 3
Factor the expression completely: \(2 y^6 - 32 y^2\).
41 Factoring - Sum/Difference of Cubes · Level 3
Factor the expression completely: \(x^6 - 1\).
42 Factoring - Grouping · Level 3
Factor the expression completely: \(y^3 - 2 y^2 - y + 2\).
43 Factoring - Rational Exponents · Level 3
Factor the expression completely: \(x^{-\dfrac{1}{2}} - 2 x^{\dfrac{1}{2}} + x^{\dfrac{3}{2}}\).
44 Factoring - Sum of Cubes · Level 3
Factor the expression completely: \(a^4 b^2 + a b^5\).
45 Factoring - Grouping · Level 3
Factor the expression completely: \(4 x^3 - 8 x^2 + 3 x - 6\).
46 Factoring - Sum of Cubes · Level 3
Factor the expression completely: \(8 x^3 + y^6\).
47 Factoring - Common Radical Factor · Level 4
Factor the expression completely: \((x^2 + 2)^{\dfrac{5}{2}} + 2 x (x^2 + 2)^{\dfrac{3}{2}} + x^2 \sqrt{x^2 + 2}\).
48 Factoring - Grouping · Level 3
Factor the expression completely: \(3 x^3 - 2 x^2 + 18 x - 12\).
49 Operations - Polynomial Multiplication · Level 2
Perform the indicated operations and simplify: \((2 x + 1)(3 x - 2) - 5(4 x - 1)\).
50 Operations - Difference of Squares · Level 1
Perform the indicated operations and simplify: \((2 y - 7)(2 y + 7)\).
51 Operations - Polynomial Subtraction · Level 2
Perform the indicated operations and simplify: \((1 + x)(2 - x) - (3 - x)(3 + x)\).
52 Operations - Radicals · Level 3
Perform the indicated operations and simplify: \(\sqrt{x} (\sqrt{x} + 1)(2 \sqrt{x} - 1)\).
53 Operations - Factoring · Level 2
Perform the indicated operations and simplify: \(x^2 (x - 2) + x (x - 2)^2\).
54 Operations - Rational Expressions · Level 3
Perform the indicated operations and simplify: \(\dfrac{x^2 - 2 x - 3}{2 x^2 + 5 x + 3}\).
55 Review - Algebraic operations · Level 2
Perform the indicated operations and simplify: \(\dfrac{x^2 - 2x - 15}{x^2 - 6x + 5} \div \dfrac{x^2 - x - 12}{x^2 - 1}\)
56 Review - Algebraic operations · Level 2
Perform the indicated operations and simplify: \(\dfrac{2}{x} + \dfrac{1}{x-2} + \dfrac{3}{(x-2)^2}\)
57 Review - Algebraic operations · Level 2
Perform the indicated operations and simplify: \(\dfrac{1}{x-1} - \dfrac{2}{x^2-1}\)
58 Review - Algebraic operations · Level 2
Perform the indicated operations and simplify: \(\dfrac{1}{x+2} + \dfrac{1}{x^2-4} - \dfrac{2}{x^2-x-2}\)
59 Review - Algebraic operations · Level 2
Simplify the complex fraction: \(\dfrac{\dfrac{1}{x} - \dfrac{1}{2}}{x - 2}\)
60 Review - Algebraic operations · Level 2
Simplify the complex fraction: \(\dfrac{\dfrac{1}{x} - \dfrac{1}{x + 1}}{\dfrac{1}{x + 1}}\)
61 Review - Rationalization · Level 2
Rationalize the denominator and simplify: \(\dfrac{\sqrt{6}}{\sqrt{3} + \sqrt{2}}\)
62 Review - Rationalization · Level 3
Rationalize the numerator: \(\dfrac{\sqrt{x+h} - \sqrt{x}}{h}\)
63 Review - Equations · Level 1
Find all real solutions of the equation: \(7x - 6 = 4x + 9\)
64 Review - Equations · Level 1
Find all real solutions of the equation: \(8 - 2x = 14 + x\)
65 Review - Equations · Level 2
Find all real solutions of the equation: \(\dfrac{x+1}{x-1} = \dfrac{3x}{3x-6}\)
66 Review - Equations · Level 2
Find all real solutions of the equation: \((x+2)^2 = (x-4)^2\)
67 Review - Equations · Level 2
Find all real solutions of the equation: \(x^2 - 9x + 14 = 0\)
68 Review - Equations · Level 2
Find all real solutions of the equation: \(x^2 + 24x + 144 = 0\)
69 Review - Equations · Level 2
Find all real solutions of the equation: \(2x^2 + x = 1\)
70 Review - Equations · Level 2
Find all real solutions of the equation: \(3x^2 + 5x - 2 = 0\)
71 Review - Equations · Level 2
Find all real solutions of the equation: \(4x^3 - 25x = 0\)
72 Review - Equations · Level 3
Find all real solutions of the equation: \(x^3 - 2x^2 - 5x + 10 = 0\)
73 Review - Equations · Level 2
Find all real solutions of the equation: \(3x^2 + 4x - 1 = 0\)
74 Review - Equations · Level 3
Find all real solutions of the equation: \(\dfrac{1}{x} + \dfrac{2}{x - 1} = 3\)
75 Review - Equations · Level 3
Find all real solutions of the equation: \(\dfrac{x}{x-2} + \dfrac{1}{x+2} = \dfrac{8}{x^2-4}\)
76 Review - Equations · Level 3
Find all real solutions of the equation: \(x^4 - 8x^2 - 9 = 0\)
77 Review - Equations · Level 2
Find all real solutions of the equation: \(|x-7| = 4\)
78 Review - Equations · Level 2
Find all real solutions of the equation: \(|2x - 5| = 9\)
79 Review - Word problems · Level 3
The owner of a store sells raisins for \$3.20 per pound and nuts for \$2.40 per pound. He decides to mix the raisins and nuts and sell 50 lb of the mixture for \$2.72 per pound. What quantities of raisins and nuts should he use?
80 Review - Word problems · Level 3
Anthony leaves Kingstown at 2:00 P.M. and drives to Queensville, 160 mi distant, at 45 mi/h. At 2:15 P.M. Helen leaves Queensville and drives to Kingstown at 40 mi/h. At what time do they pass each other on the road?
81 Review - Word problems · Level 3
A woman cycles 8 mi/h faster than she runs. Every morning she cycles 4 mi and runs \(2 \dfrac{1}{2}\) mi, for a total of one hour of exercise. How fast does she run?
82 Review - Word problems · Level 3
The hypotenuse of a right triangle has length 20 cm. The sum of the lengths of the other two sides is 28 cm. Find the lengths of the other two sides of the triangle.
83 Review - Word problems · Level 3
Abbie paints twice as fast as Beth and three times as fast as Cathie. If it takes them 60 min to paint a living room with all three working together, how long would it take Abbie if she works alone?
84 Review - Word problems · Level 3
A homeowner wishes to fence in three adjoining garden plots, one for each of her children, as shown in the figure. If each plot is to be 80 ft\(^2\) in area and she has 88 ft of fencing material at hand, what dimensions should each plot have?
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85 Review - Inequalities · Level 2
Solve the inequality. Express the solution using interval notation and graph the solution set on the real number line: \(3x - 2 > -11\)
86 Review - Inequalities · Level 2
Solve the inequality. Express the solution using interval notation and graph the solution set on the real number line: \(-1 < 2x + 5 \leq 3\)
87 Review - Inequalities · Level 2
Solve the inequality. Express the solution using interval notation and graph the solution set on the real number line: \(x^2 + 4x - 12 > 0\)
88 Review - Inequalities · Level 2
Solve the inequality. Express the solution using interval notation and graph the solution set on the real number line: \(x^2 \leq 1\)
89 Review - Inequalities · Level 3
Solve the inequality. Express the solution using interval notation and graph the solution set on the real number line: \(\dfrac{x-4}{x^2-4} \leq 0\)
90 Review - Inequalities · Level 3
Solve the inequality. Express the solution using interval notation and graph the solution set on the real number line: \(\dfrac{5}{x^3-x^2-4x+4} < 0\)
91 Review - Inequalities · Level 2
Solve the inequality. Express the solution using interval notation and graph the solution set on the real number line: \(|x-5| \leq 3\)
92 Review - Inequalities · Level 2
Solve the inequality. Express the solution using interval notation and graph the solution set on the real number line: \(|x-4| < 0.02\)
93 Review - Graphical solutions · Level 3
Use a graphing device to solve the equation: \(x^2 - 4x = 2x + 7\)
94 Review - Graphical solutions · Level 3
Use a graphing device to solve the equation: \(\sqrt{x+4} = x^2 - 5\)
95 Review - Graphical solutions · Level 3
Use a graphing device to solve the inequality: \(4x - 3 \geq x^2\)
96 Review - Graphical solutions · Level 3
Use a graphing device to solve the inequality: \(x^3 - 4x^2 - 5x > 2\)
97 Review - Coordinate geometry · Level 3
Two points \(P\) and \(Q\) are given: \(P(2, 0)\), \(Q(-5, 12)\). (a) Plot \(P\) and \(Q\) on a coordinate plane. (b) Find the distance from \(P\) to \(Q\). (c) Find the midpoint of the segment \(P Q\). (d) Sketch the line determined by \(P\) and \(Q\), and find its equation in slope-intercept form. (e) Sketch the circle that passes through \(Q\) and has center \(P\), and find the equation of this circle.
98 Review - Coordinate geometry · Level 3
Two points \(P\) and \(Q\) are given: \(P(7, -1)\), \(Q(2, -11)\). (a) Plot \(P\) and \(Q\) on a coordinate plane. (b) Find the distance from \(P\) to \(Q\). (c) Find the midpoint of the segment \(P Q\). (d) Sketch the line determined by \(P\) and \(Q\), and find its equation in slope-intercept form. (e) Sketch the circle that passes through \(Q\) and has center \(P\), and find the equation of this circle.
99 Review - Coordinate geometry · Level 2
Sketch the region given by the set of all points \((x, y)\) such that \(-4 < x < 4\) and \(-2 < y < 2\).
100 Review - Coordinate geometry · Level 2
Sketch the region given by the set of all points \((x, y)\) such that \(x \geq 4\) or \(y \geq 2\).
101 Review - Coordinate geometry · Level 2
Which of the points \(A(4, 4)\) or \(B(5, 3)\) is closer to the point \(C(-1, -3)\)?
102 Review - Circles · Level 2
Find an equation of the circle that has center \((2, -5)\) and radius (radius value missing from OCR).
103 Review - Circles · Level 2
Find an equation of the circle that has center \((-5, -1)\) and passes through the origin.
104 Review - Circles · Level 3
Find an equation of the circle that contains the points \(P(2, 3)\) and \(Q(-1, 8)\) and has the midpoint of the segment \(P Q\) as its center.
105 Review - Circles · Level 3
Determine whether the equation represents a circle, represents a point, or has no graph. If the equation is that of a circle, find its center and radius: \(x^2 + y^2 + 2x - 6y + 9 = 0\)
106 Review - Circles · Level 3
Determine whether the equation represents a circle, represents a point, or has no graph. If the equation is that of a circle, find its center and radius: \(2x^2 + 2y^2 - 2x + 8y = \dfrac{1}{2}\)
107 Review - Circles · Level 3
Determine whether the equation represents a circle, represents a point, or has no graph. If the equation is that of a circle, find its center and radius: \(x^2 + y^2 + 72 = 12x\)
108 Review - Circles · Level 3
Determine whether the equation represents a circle, represents a point, or has no graph. If the equation is that of a circle, find its center and radius: \(x^2 + y^2 - 6x - 10y + 34 = 0\)
109 Review - Symmetry and graphs · Level 2
Test the equation for symmetry, and sketch its graph: \(y = 2 - 3x\)
110 Review - Symmetry and graphs · Level 2
Test the equation for symmetry, and sketch its graph: \(2x - y + 1 = 0\)
111 Review - Symmetry and graphs · Level 2
Test the equation for symmetry, and sketch its graph: \(x + 3y = 21\)
112 Review - Symmetry and graphs · Level 2
Test the equation for symmetry, and sketch its graph: \(x = 2y + 12\)
113 Review - Symmetry and graphs · Level 2
Test the equation for symmetry, and sketch its graph: \(y = 16 - x^2\)
114 Review - Symmetry and graphs · Level 2
Test the equation for symmetry, and sketch its graph: \(8x + y^2 = 0\)
115 Review - Symmetry and graphs · Level 2
Test the equation for symmetry, and sketch its graph: \(x = \sqrt{y}\)
116 Review - Symmetry and graphs · Level 2
Test the equation for symmetry, and sketch its graph: \(y = -\sqrt{1 - x^2}\)
117 Review - Graphing technology · Level 2
Use a graphing device to graph the equation in an appropriate viewing rectangle: \(y = x^2 - 6x\)
118 Review - Graphing technology · Level 2
Use a graphing device to graph the equation in an appropriate viewing rectangle: \(y = \sqrt{5 - x}\)
119 Review - Graphing technology · Level 2
Use a graphing device to graph the equation in an appropriate viewing rectangle: \(y = x^3 - 4x^2 - 5x\)
120 Review - Graphing technology · Level 2
Use a graphing device to graph the equation in an appropriate viewing rectangle: \(\dfrac{x^2}{4} + y^2 = 1\)
121 Review - Lines · Level 2
Find an equation for the line that passes through the points \((-1, -6)\) and \((2, -4)\).
122 Review - Lines · Level 2
Find an equation for the line that passes through the point \((6, -3)\) and has slope \(-\dfrac{1}{2}\).
123 Review - Lines · Level 3
Find an equation for the line that passes through the point \((1, 7)\) and is perpendicular to the line \(x - 3y + 16 = 0\).
124 Review - Lines · Level 3
Find an equation for the line that passes through the origin and is parallel to the line \(3x + 15y = 22\).
125 Review - Lines · Level 3
Find an equation for the line that passes through the point \((5, 2)\) and is parallel to the line passing through \((-1, -3)\) and \((3, 2)\).
126 Review - Equations from figures · Level 3
Find equations for the circle and the line in the figure.
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127 Review - Equations from figures · Level 3
Find equations for the circle and the line in the figure.
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128 Review - Variation and applications · Level 3
Hooke's Law states that if a weight \(w\) is attached to a hanging spring, then the stretched length \(s\) of the spring is linearly related to \(w\). For a particular spring we have \(s = 0.3w + 2.5\) where \(s\) is measured in inches and \(w\) in pounds. (a) What do the slope and \(s\)-intercept in this equation represent? (b) How long is the spring when a 5-lb weight is attached?
129 Review - Variation and applications · Level 3
Margarita is hired by an accounting firm at a salary of \$60,000 per year. Three years later her annual salary has increased to \$70,500. Assume that her salary increases linearly. (a) Find an equation that relates her annual salary \(S\) and the number of years \(t\) that she has worked for the firm. (b) What do the slope and \(S\)-intercept of her salary equation represent? (c) What will her salary be after 12 years with the firm?
130 Review - Variation and applications · Level 2
Suppose that \(M\) varies directly as \(z\), and \(M = 120\) when \(z = 15\). Write an equation that expresses this variation.
131 Review - Variation and applications · Level 2
Suppose that \(z\) is inversely proportional to \(y\), and that \(z = 12\) when \(y = 16\). Write an equation that expresses \(z\) in terms of \(y\).
132 Review - Variation and applications · Level 3
The intensity of illumination \(I\) from a light varies inversely as the square of the distance \(d\) from the light. (a) Write this statement as an equation. (b) Determine the constant of proportionality if it is known that a lamp has an intensity of 1000 candles at a distance of 8 m. (c) What is the intensity of this lamp at a distance of 20 m?
133 Review - Variation and applications · Level 3
The frequency of a vibrating string under constant tension is inversely proportional to its length. If a violin string 12 inches long vibrates 440 times per second, to what length must it be shortened to vibrate 660 times per second?
134 Review - Variation and applications · Level 3
The terminal velocity of a parachutist is directly proportional to the square root of his weight. A 160-lb parachutist attains a terminal velocity of 9 mi/h. What is the terminal velocity for a parachutist weighing 240 lb?
135 Review - Variation and applications · Level 3
The maximum range of a projectile is directly proportional to the square of its velocity. A baseball pitcher throws a ball at 60 mi/h, with a maximum range of 242 ft. What is his maximum range if he throws the ball at 70 mi/h?
136 Concept Check - Vertical and Horizontal Lines · Level 1
(a) What is the equation of a vertical line?
(b) What is the equation of a horizontal line?

Enter your answer directly below each part above.

137 Concept Check - General Equation of a Line · Level 1
What is the general equation of a line?
138 Concept Check - Parallel and Perpendicular Lines · Level 2
Given lines with slopes \(m_1\) and \(m_2\), explain how you can tell if the lines are
(a) parallel
(b) perpendicular

Enter your answer directly below each part above.

139 Concept Check - Variation · Level 1
Write an equation that expresses each relationship.
(a) \(y\) is directly proportional to \(x\).
(b) \(y\) is inversely proportional to \(x\).
(c) \(z\) is jointly proportional to \(x\) and \(y\).

Enter your answer directly below each part above.

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