Rising Geometry Students-Summer Math Skills

Page 1

Simplify the following expressions.

1. 3( x + 1) + 5 2. (x + 2)(x + 5) 3. -2(-3)

Find the slope of the line through the given points.

4. ( -1, 3) and ( 2, 7) 5. ( -2, 5) and (1, -1)

6. Write each as the product of two factors.

a. x² + 6x + 8 b. x² - x - 12 c. x² - 25 d. x² + 8x + 16

7. Simplify.

a. 3 + 9 • 7 – 8 ÷ 2 b. ( −2 )5 c. −25 d. -21 – 2 • 6 ÷ 3

8. Employee wages are directly proportional to time worked. If Liza received $ 114 for 20

hours work, how much should she receive for 30 hours work?

9. If a = -3, c = -1, and d = 2, find the value of 𝑐−7𝑑

𝑎−𝑑 .

For problems 10-12, solve for x.

10. |x| + 2 = 3 11. |x - 2| = 3 12. |x| + 6 = 2

Page 2

1. Evaluate the following

a. √25 b. √−25 c. | 8 |

d. | -8 | e. - | 8 | f. - | -8 |

2. Multiply the following.

a. (2x – 3)(x + 1) b. ( x – 7 )² c. ( x – 6 )( x + 6 )

3. The length of a rectangular lot is twice the width. The perimeter is 156 meters.

Write and equation and show all work to find the width and length.

For problems 4-6, solve for x, showing all work.

4. X – 7 – 4x + 4 = 0 5. 2 ( x – 1 ) – 7 = 5 6. 7 – 4 ( x + 1 ) = 15

7. Solve, and graph on a number line.

a. | x – 2 | ˂ 7 b. | x | ≥ 4 c. | 9 – x | = 2 d. | x | = -5

8. Find the x- and y- intercepts for the line 3x + 5y = 15.

For problems 9-10, simplify.

9. 1

2 ( 5x – 2 ) – 5 (

1

2 x – 2 ) 10. 2x ( x + 3 ) – ( x² + 6x )

Page 3

1. In right triangle ABC, the hypotenuse c = 11 and the leg a = 7. Find the length of side b

in simplified radical form. Round side b to the nearest hundredth.

2. Simplify

a. √48 b. √50𝑥² c. √72𝑦³ d. √75𝑥4𝑦9

3. Find the area of the shaded region.

Leave answer in factored form.

4. Solve.

a. 8( 2 – x ) = 4( 4 – 2x ) b. 5( 3 – 2y ) = 10( 2- y )

Solve for x, showing all work.

5. x² - 2 = 7 6. ( x – 2 )² = 9

7. The points (8,3) and (5,y) are on the same line. If the slope of the line is 1

2 , find y.

8. Simplify.

a. -8m(−2𝑚3)4 b. ( 3x )² • ( 4x² )³ c. ( 1 - 3² ) ÷ ( 1 -3 )² - (-20)

9. Graph on one coordinate graph. y ˂ 2x – 4 and x + 2y ≤ 7

10. Find two numbers whose sum is 12 and whose product is 20.

Page 4

1. Find a number between 1

4 and

1

5 .

2. The sides of a triangle are consecutive odd integers. If the perimeter of the triangle is

141 cm, find the length of the largest side.

3. Use the quadratic formula to solve 3x² + x – 2 = 0 for x.

Complete 4-6.

4. Find the slope of the line with x-intercept of 3 and y-intercept of 5.

5. Write the equation of this line.

6. Prove whether ( 6, -5 ) is on this line. Show work algebraically.

7. Three sides of a triangle are in the ratio of 2:3:4. Its perimeter is 72 units. Find the lengths

of all three sides.

8. Find the equation of the line perpendicular to y = -2x + 4, and containing ( -4, 5 ).

9. Simplify (2𝑥−2𝑦3)

−2

(2𝑥3𝑦−1)−3

10. Identify the zero(s) of the following function: f(x) = 2x2 – x – 6

Page 5

For problems 1-3, solve for x.

1. x² + 2x² = 147 2. x² - 6x = 4 3. √𝑥 = -5

4. Solve the following system using substitution. X = y - 2

-2x + 3y = 8

3. Find the rule that describes this relationship. ( 0, 3 ), ( 1, 5 ), ( 4, 11 ), ( 6, 15 )

Use ∆ABC at the right.

4. Find the length of AC.

5. Find the length of BC.

6. Find the length of AB.

7. Verify that ∆ABC is a right triangle.

8. Find the slope of AB.

9. Find the perimeter of ∆ABC.

10. Find the area of ∆ABC.

11. Simplify.

a. 4𝑥2

𝑥2−1 •

𝑥+1

6𝑥 b.

1

6𝑥2 - 5

4𝑥 c.

5𝑥+5

𝑥3+𝑥2 ÷ 2

𝑥2 d. 𝑥

𝑥−𝑦 +

𝑦

𝑦−𝑥

A(1,5)

B(1, 1)

C(4, 1)

Page 6

1. Find the equation of the line with slope - 1

2 that goes through point ( -2, 5 ).

2. The braking distance needed to stop a car is directly proportional to the square of the

car’s speed. If it takes 55 yards to stop a car traveling at 50 mph, how many yards will it

take to stop a car traveling at 65 mph?

3. Find all the values of k for which the trinomial x² + kx + 24 can be factored.

4. Can 1

3,

1

4, and

5

12 be the sides of a right triangle? Justify your answer.

The points ( 2, 4 ) and ( -2, 8 ) lie on a line.

5. Find the slope of this line.

6. Find the y-intercept of this line.

7. Write the equation of this line.

Solve x² - 2x – 3 = 5.

8. Use the quadratic formula.

9. Use completing the square.

10. Use factoring.

Which method do you like?

11. Graph the parabola f(x) = x ² + 4x + 3. Find

a. equation of the axis of symmetry

b. the vertex, and determine if it is a maximum or

minimum

c. the y-intercept

d. the domain and range of the function

12. Perform the indicated operations.

a. −3𝑥

𝑦² •

𝑥𝑦6

18 b.

2𝑥

3 +

3

4𝑥 c. m +

2

𝑚−1 d.

( 𝑥−5 )³

3𝑥³ ÷

5−𝑥

9𝑥

Page 7

For 1-3, simplify.

1. 2𝑎

2𝑐 2. (2𝑥2𝑦)−3 3.

2

𝑐2 ÷

3

𝑐

4. Factor completely.

a. 2𝜋rh + 𝜋r² b. 3x² - 19x + 20 c. 36x - x³

5. Solve these systems of equations by any method.

a. 5x + y = 23 b. 5x + 4y = 14 c. 2x + y = 4 d. 2x + 3y = 3

5x – 6y = 2 5x = 3y + 7 5x + 2y = 0 4x + 3y = 3

6. Graph on a number line.

a. 7 – 3y ˂ 9 b. 5 ˂ 2h – 1 ≤ 9 c. 2x + 1 ˂ 5 and x ˃ 1 d. p ≥ 3 or p ˂ -5

For problems 7-9, simplify. Use positive exponents in the answers.

7. 2𝑥

3 -

𝑥−2

4 8.

1

2 x(4x + 2 ) +

1

3 (6x² ) 9.

2

𝑥2 •

3𝑥3

4

10. Mr. Harris leaves on a bike trip pedaling at an average rate of 16 mph. After he has

been gone for 1 hour, his wife leaves in her car to meet him for a picnic lunch. If she drives

at an average rate of 40 mph, how long will it take her to meet with him?

Page 8

1. Complete to make a perfect square trinomial.

a. x² + 10x + ____ b. x² - 24x + ____ c. x² + x + ____

2. Given f(x) = x² - 2x + 1 and g(x) = 2x, find

a. f(0) b. f(-1) c. g(0)

d. g(-1) e. f(g(2)) f. g(f(2))

3. Jim and Bob are throwing a ball into the air. The height of the ball as a function of

time (in seconds) is h(t) = -5t² + 20t + 2.

What is the height of the ball 3 seconds after Jim threw the ball in the air?

4. Simplify, leaving your answer in scientific notation.

a. 2 × 10−3

8 × 10−7 b. ( 8 x 10−2)( 2 x 107)

5. Find the rate of change.

6. Factor completely.

a. 5x² + 17x – 12 b. 3x² - 15x – 18 c. 3x(y + 1) – 7(y + 1)

7. Solve for x

a. √𝑥 = 5 b. √𝑥 + √𝑥 = 32 c. √2𝑥 - 9 = 1

8. What is the solution of this system of equations?

4x – y = 10

2x – 3y = 0

9. Solve for m.

a. 6m – 5p = h b. 2ℎ−5

𝑚 = p

10. List restrictions for the variable. Simplify.

a. 𝑥²−4𝑥−5

𝑥²−1 b.

𝑥²+7𝑥+10

𝑥²+3𝑥+2 c.

𝑥²−𝑥−56

𝑥²+𝑥−42

11. Graph 1

3x -

1

2 y = 1

12. What are the intercepts of

the equation above?

Page 9

1. Evaluate when a = -9 and c = 1

3 .

a. ac b. 𝑎

𝑐 c. 𝑐−2 d. - a² e.

−𝑐

6

2. A horizontal line has a slope of ________ and a vertical line has a slope of ________.

Rationalize all denominators.

3. √5

6 4.

1

3− √3 5.

√3

√8 6.

√10

3√15

7. A line passes through ( -3,4 ) and has a slope of −1

2 . Name two more points on this line.

8. The sum of two numbers is 45, and their difference is 9. Find the numbers.

9. A perfectly square lot has a perimeter of exactly one mile. What is its area?

Use the points ( -2, 5 ) and ( 4, 13 ) for 10-11.

10. Find the slope of the line containing these points.

11. Find the distance between these two points.

Page 10

1. Evaluate when a= -3, c = 1

3 , and d = -9.

a. a³ b. – d – a c. 𝑎

𝑐 d. √𝑐 e. acd

2. A line has a slope of 2. One point on the line is ( -2, 3 ). Another is ( x, -3 ). Find x.

Solve the systems of equations, using all or no solutions if needed.

3. 3x + 4y = 11 4. Y = 2x + 3

4x + 3y = 10 y – 2x = 3

5. Graph the solution of 17 – 5x ≥ 2x – 7 – 3x.

6. Simplify.

a.

2𝑥

3𝑥²

6

b. 4

1

2

11

4

c.

2𝑥−6

𝑥+1𝑥²−9

3𝑥²−3

7. Name the inequalities shown by the

graph.

8. Label the property that justifies each step.

a. - x + [−𝑦 + (𝑦 + 𝑥)] = - x + [(−𝑦 + 𝑦) + 𝑥]

b. = -x + [𝑜 + 𝑥]

c. = -x + [𝑥]

d. = 0

9. One angle in a triangle is twice another. The third angle is 10 less than the larger angle. Find

all three angles.

10. Given the parabola f(x) = x² + 3x.

a. Find the vertex

b. Find the axis of symmetry

c. Complete 5 points in order to graph.

d. Identify the domain and range of the

function

11. Simplify, assuming no zero denominators.

a. 4𝑥²𝑦

5𝑥³ •

15𝑥

14𝑦 b.

𝑥²−16

2 •

3𝑥

𝑥−4 c.

3𝑥

𝑥²−9 ÷

6𝑥²

𝑥²−𝑥−6

12. Solve these quadratics by any method you choose.

a. ( x – 1 )² = 16 b. x² - 8x = 2 c. x² + 2x = 1

d. 2y² = 26y – 24 e. 2x² + 9 = 0 f. x² - 4x = 5