Download Rising Geometry Students-Summer Math Skills Page 1 Simplify the

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 |
b. ( x – 7 )²
c. ( x – 6 )( x + 6 )
2. Multiply the following.
a. (2x – 3)(x + 1)
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
8. Find the x- and y- intercepts for the line 3x + 5y = 15.
For problems 9-10, simplify.
9.
1
2
1
( 5x – 2 ) – 5 ( 2 x – 2 )
10. 2x ( x + 3 ) – ( x² + 6x )
d. | x | = -5
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. √75𝑥 4 𝑦 9
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 −2 𝑦 3 )
−2
(2𝑥 3 𝑦 −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
3. √𝑥 = -5
2. x² - 6x = 4
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.
A(1,5)
4. Find the length of AC.
5. Find the length of BC.
C(4, 1)
6. Find the length of AB.
B(1, 1)
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.
4𝑥 2
𝑥 2 −1
•
𝑥+1
6𝑥
b.
1
6𝑥 2
-
5
4𝑥
c.
5𝑥+5
𝑥 3 +𝑥 2
÷
2
𝑥2
d.
𝑥
𝑥−𝑦
+
𝑦
𝑦−𝑥
Page 6
1.
2.
1
Find the equation of the line with slope - 2 that goes through point ( -2, 5 ).
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
1
5
3
4
12
,
, and
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
•
𝑥𝑦 6
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
÷
2. (2𝑥 2 𝑦)−3
3.
b. 3x² - 19x + 20
c. 36x - x³
𝑐2
3
𝑐
4. Factor completely.
a. 2𝜋rh + 𝜋r²
5. Solve these systems of equations by any method.
a. 5x + y = 23
b. 5x + 4y = 14
5x – 6y = 2
c. 2x + y = 4
5x = 3y + 7
d. 2x + 3y = 3
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
x(4x + 2 ) +
2
1
(6x² )
3
9.
2
𝑥2
•
3𝑥 3
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
b.
𝑥²−1
11. Graph
1
1
3
2
x- y=1
12. What are the intercepts of
the equation above?
𝑥²+7𝑥+10
𝑥²+3𝑥+2
c.
𝑥²−𝑥−56
𝑥²+𝑥−42
Page 9
1
1. Evaluate when a = -9 and c = 3 .
a. ac
b.
𝑎
c. 𝑐 −2
𝑐
d. - a²
e.
−𝑐
6
2. A horizontal line has a slope of ________ and a vertical line has a slope of ________.
Rationalize all denominators.
5
3. √6
4.
1
3− √3
5.
√3
6.
√8
7. A line passes through ( -3,4 ) and has a slope of
−1
2
√10
3√15
. 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
1. Evaluate when a= -3, c = 3 , and d = -9.
a. a³
b. – d – a
c.
𝑎
d. √𝑐
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
3
𝑥²
6
1
b.
42
1
1
4
7. Name the inequalities shown by the
graph.
c.
2𝑥−6
𝑥+1
𝑥²−9
3𝑥²−3
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