Math Diagnostic 3: Pre-Algebra to Analytic Geometry

Family 10 · Exponents, roots, and scientific notation

Exponents, roots, and scientific notation

1.Write as a single power: 2³ × 2⁴
2.Simplify: (x³)²
3.Write 3⁻² as a fraction.
4.Write 4,500,000 in scientific notation.

Family 11 · Expressions and polynomials

Expressions and polynomials

5.Simplify: 3x + 5 − x + 2
6.Multiply out: 2(x + 4)
7.Simplify: (4x + 1) − (x − 2)
8.Multiply out: (x + 2)(x + 3)
9.Solve: x² − 5x + 6 = 0

Family 12 · Equations and inequalities

Equations and inequalities

10.Solve for x: 3x + 4 = 19
11.Solve for x: 5x − 2 = 3x + 8
12.Solve: 2x − 3 < 7
13.Solve for x: x4 = 912

Family 12 · Equations and inequalities

Algebra word problems

14.A plumber charges $50 to come out, plus $30 an hour. A job cost $170. How many hours did it take?
15.Three consecutive whole numbers add up to 45. What is the smallest of them?
16.A rectangle is 3 feet longer than it is wide. Its perimeter is 26 feet. How wide is it?

Family 13 · Lines, slope, and systems

Linear equations and slope

17.What is the slope of the line through (1, 2) and (4, 8)?
18.For the line y = 3x − 5, what are the slope and the y-intercept? slope y-intercept
19.Write the equation of the line with slope 2 that crosses the y-axis at (0, 3).
20.The line passes through the two marked points. What is its slope?

Family 13 · Lines, slope, and systems

Systems of equations

21.Solve the system: x + y = 10 and x − y = 4 x y
22.At a movie theater, adult tickets cost $8 and child tickets cost $5. 20 tickets were sold for $145 in all. How many of each? adult child

Family 7 · Geometry

Geometry

23.A triangle has a base of 10 inches and a height of 6 inches. What is its area?
24.A circle has a radius of 5 cm. What is its area? Use 3.14 for π.
25.A right triangle has legs of 6 and 8. How long is the hypotenuse?
26.Two angles together make a straight line. One is 110°. What is the other?
27.A 6-foot-tall person casts a 4-foot shadow. At the same time a tree casts a 20-foot shadow. How tall is the tree?

Family 14 · Functions and the coordinate plane

Coordinate geometry

28.What is the distance between A (1, 1) and B (4, 5)?
29.What is the midpoint of the segment from A (2, 3) to B (6, 7)?
30.Where does the line y = −x + 4 cross the x-axis?

Family 14 · Functions and the coordinate plane

Functions

31.If f(x) = 2x + 1, what is f(3)?
32.
x y
1 3
2 5
3 7
4 9
Which rule fits the table?

x	y
1	3
2	5
3	7
4	9

Or check one family at a time, with the button under it.

Results

Where to go next

These nine sets sit inside six of the fourteen families. Two of them look at family 12, two at family 13, and two at family 14, because that is where the test asks the most.

Family	Questions	Right	Where to go next
Family 10 · Exponents, roots, and scientific notation	1–4	of 4	Go to Family 10Creatures in Flight, Quizzes 29–32 and 31a; The Exponent Machine; The Rules, One Page
Family 11 · Expressions and polynomials	5–9	of 5	Go to Family 11Like With Like, room one of the polynomials series
Family 12 · Equations and inequalities	10–13	of 4	Go to Family 12The Thing Called x, The Balance, The Share, and Algebra Practice 1
Family 12 · Equations and inequalitiesAlgebra word problems	14–16	of 3	Go to Family 12Algebra Practice 1 (words into symbols), and the word-problem set in Lines on the Grid
Family 13 · Lines, slope, and systemsLinear equations and slope	17–20	of 4	Go to Family 13Lines on the Grid — slope from a graph, from two points, and from a table
Family 13 · Lines, slope, and systemsSystems of equations	21–22	of 2	Go to Family 13Lines on the Grid — systems by substitution and by elimination
Family 7 · Geometry	23–27	of 5	Go to Family 7Quiz 27 for perimeter and area; Circles and Equations; The Third Side, on paper
Family 14 · Functions and the coordinate planeCoordinate geometry	28–30	of 3	Go to Family 14Lines on the Grid — the grid, distance, and the midpoint
Family 14 · Functions and the coordinate planeFunctions	31–32	of 2	Go to Family 14Lines on the Grid — function notation, and working a rule on a number

These nine sections are the closer look at six of the fourteen families on the Flyover: geometry; exponents, roots, and scientific notation; expressions and polynomials; equations and inequalities; lines, slope, and systems; and functions and the coordinate plane. To check every family at once, take the Flyover Diagnostic.

The quiz series is at thepeoplesshare.org/ged/math.

The key prints only when it is showing, and it starts on its own page. After you check a topic, each question you missed offers its own reasoning with the Why? button.

Answer key

Answers, with the reasoning in brief

1.2⁷, which is 128. Same base, so keep the base and add the exponents: 3 + 4 = 7. Written out, that is three 2s times four 2s, seven 2s in all.
2.x⁶. A power raised to a power: multiply the exponents. 3 × 2 = 6. (It is x³ written twice and multiplied: six factors of x.)
3.19. A negative exponent means "one over": 3⁻² = 1/3² = 1/9.
4.4.5 × 10⁶. Move the decimal point until one digit stands before it: 4.5. It moved 6 places, so the power of ten is 6.
5.2x + 7. Gather like with like. The x terms: 3x − x = 2x. The plain numbers: 5 + 2 = 7.
6.2x + 8. Multiply each term inside by 2: 2 × x = 2x and 2 × 4 = 8.
7.3x + 3. Subtracting the parentheses changes the sign of every term inside: −(x − 2) = −x + 2. Then gather: 4x − x = 3x, and 1 + 2 = 3.
8.x² + 5x + 6. Multiply every term in the first parentheses by every term in the second: x × x = x², x × 3 = 3x, 2 × x = 2x, 2 × 3 = 6. Gather the x terms: 3x + 2x = 5x.
9.x = 2 or x = 3. Find two numbers that multiply to 6 and add to 5: 2 and 3. So x² − 5x + 6 = (x − 2)(x − 3), and a product is 0 only when a factor is 0.
10.x = 5. Undo the addition first: take 4 from both sides, 3x = 15. Then undo the multiplication: divide by 3, x = 5. Check: 3 × 5 + 4 = 19.
11.x = 5. Gather the x terms on one side: take 3x from both sides, 2x − 2 = 8. Add 2 to both sides: 2x = 10. Divide by 2: x = 5.
12.x < 5. Solve it like an equation: add 3 to both sides, 2x < 10, then divide by 2, x < 5. The sign stays as it is because you divided by a positive number.
13.x = 3. 12 is 3 times 4, so x must be 9 ÷ 3 = 3. Or cross-multiply: 12x = 36, so x = 3.
14.4 hours. Let h be the hours: 50 + 30h = 170. Take 50 from both sides: 30h = 120. Divide by 30: h = 4.
15.14. Call the smallest n. The next two are n + 1 and n + 2, so n + (n + 1) + (n + 2) = 3n + 3 = 45. Then 3n = 42 and n = 14. Check: 14 + 15 + 16 = 45.
16.5 feet. Let w be the width; the length is w + 3. Perimeter: 2w + 2(w + 3) = 4w + 6 = 26. So 4w = 20 and w = 5. Check: 5 + 8 + 5 + 8 = 26.
17.2. Slope is rise over run. Rise: 8 − 2 = 6. Run: 4 − 1 = 3. Slope = 6/3 = 2.
18.Slope 3, y-intercept −5. In y = mx + b, the number multiplying x is the slope and the number added is the y-intercept, where the line crosses the y-axis.
19.y = 2x + 3. Put the slope and the y-intercept into y = mx + b.
20.2. Read the two marked points: (0, 1) and (2, 5). Rise: 4. Run: 2. Slope = 4/2 = 2.
21.x = 7, y = 3. Add the two equations and the y terms cancel: 2x = 14, so x = 7. Then 7 + y = 10, so y = 3.
22.15 adult, 5 child. Let a be adult tickets and c child tickets: a + c = 20 and 8a + 5c = 145. From the first, c = 20 − a. Put that into the second: 8a + 5(20 − a) = 145, so 3a + 100 = 145, 3a = 45, a = 15. Then c = 5.
23.30 square inches. Area of a triangle is half of base × height: 10 × 6 = 60, and half of that is 30.
24.78.5 square cm. Area of a circle is π × r²: 3.14 × 5 × 5 = 3.14 × 25 = 78.5. (Exactly, 25π.)
25.10. Pythagorean theorem: a² + b² = c². 6² + 8² = 36 + 64 = 100, and √100 = 10.
26.70°. Angles along a straight line add to 180°: 180 − 110 = 70.
27.30 feet. The sun makes the two triangles the same shape, so height and shadow are in proportion: 6/4 = h/20. The shadow is 5 times longer (4 × 5 = 20), so the height is 5 times taller: 6 × 5 = 30.
28.5. Draw the right triangle between the points: across 3 and up 4. The distance is the hypotenuse: √(3² + 4²) = √25 = 5.
29.(4, 5). The midpoint averages the x values and averages the y values: (2 + 6) ÷ 2 = 4 and (3 + 7) ÷ 2 = 5.
30.(4, 0). On the x-axis, y = 0. Set y to 0: 0 = −x + 4, so x = 4.
31.7. f(3) means put 3 in place of x: 2 × 3 + 1 = 7.
32.y = 2x + 1. Test each rule on the first row, x = 1: only y = 2x + 1 gives 3. Check another row: x = 2 gives 5. Each step of 1 in x adds 2 to y, which is the slope.