# Preguntas de Ã¡lgebra con respuestas y soluciones para el grado 8

Se presentan preguntas de Ã¡lgebra de octavo grado con soluciones. Se incluyen preguntas sobre cÃ³mo resolver ecuaciones, simplificar expresiones, incluidas expresiones con fracciones.

NOTE: In what follows, mixed numbers are written in the form a b/c. For example 2 1/3 means the mixed number 2 + 1/3.

1. Simplify the following algebraic expressions.
A) -2x + 5 + 10x - 9
B) 3(x + 7) + 2(-x + 4) + 5x

2. Simplify the expressions.
A) (2x - 6) / 2
B) (-x - 2) / (x + 2)
C) (5x - 5)/10

3. Solve for x the following equations.
A) -x = 6
B) 2x - 8 = -x + 4
C) 2x + 1/2 = 2/3
D) x/3 + 2 = 5
E) -5/x = 2

4. Evaluate for the given values of x and y.
A) x2 - y2 , for x = 4 and y = 5
B) |4x - 2y| , for x = -2 and y = 3
C) 3x3 - 4y4 , for x = -1 and y = -2

5. Solve the following inequalities.
A) x + 6 < 0
B) x + 1 > 5
C) 2(x - 2) < 12

6. What is the reciprocal of each of the following numbers?
A) -1
B) 0
C) 3/4
D) 2 5/7
E) 0.02

7. Evaluate the following expressions involving mixed numbers.
A) 3 3/4 + 6 1/7
B) (1 3/5) × (3 1/3) - 2 1/2
C) (5 2/3) ÷ (4 1/5)
D) (3 4/7 - 1 1/2) ÷ (2 3/8 + 2 1/4)

8. Evaluate the folwoing exponential expressions.
A) -42
B) (-2)3
C) (-2)4
D) 10000
E) 5661

9. Convert to fractions and write in simplest form.
A) 0.02
B) 12%
C) 0.5%
D) 1.12

10. Convert to decimals.
A) 1/5
B) 120%
C) 0.2%
D) 4 8/5

11. Convert to percent.
A) 3/10
B) 1.4
C) 123.45
D) 2 4/5

12. Which of these numbers is divisible by 3?
A) 156312
B) 176314

13. Which of these numbers is divisible by 4?
A) 3432
B) 1257

14. Which of these numbers is divisible by 6?
A) 1233
B) 3432

15. Which of these numbers is divisible by 9?
A) 2538
B) 1451

16. Evaluate 8x + 7 given that x - 3 = 10.

Solutions and Answers to the Above Questions

1. A) -2x + 5 + 10x - 9 : given
= (10x - 2x) + (5 - 9) : put like terms together
= 8x - 4 : group

B) 3(x + 7) + 2(-x + 4) + 5x : given
= 3x + 21 - 2x + 8 + 5x : expand
= (3x - 2x + 5x) + (21 + 8) : put like terms together
= 6x + 29 : group

2. A) (2x - 6) / 2 : given
= 2(x - 3) / 2 : factor 2 in numerator
= x - 3 : divide numerator and denominator by 2 to simplify

B) (-x - 2) / (x + 2) : given
= -1(x + 2) / (x + 2) : factor -1 in numerator
= -1 : divide numerator and denominator by x + 2 to simplify

C) (5x - 5)/10 : given
= 5(x - 1) / 10 : factor 5 in numerator
= (x - 1) / 2 : divide numerator and denominator by 5 to simplify

3. A) -x = 6 : given
x = -6 : multiply both sides of the equation by -1

B) 2x - 8 = -x + 4 : given
2x - 8 + 8 = -x + 4 + 8 : add +8 to both sides of the equation
2x = -x + 12 : group like terms
2x + x = -x + 12 + x : add +x to both sides
3x = 12 : group like terms
x = 4 : mutliply both sides by 1/3

C) 2x + 1/2 = 2/3 : given
2x + 1/2 - 1/2 = 2/3 - 1/2 : subtract 1/2 from both sides
2x = 1/6 : group like terms
x = 1/12 : multiply both sides by 1/2

D) x/3 + 2 = 5 : given
x/3 + 2 - 2 = 5 - 2 : subtract 2 from both sides
x/3 = 3 : group like terms
x = 9 : multiply both sides by 1/2

E) -5/x = 2 : given
-5 = 2x : multiply both sides by x and simplify
-5/2 = x : : multiply both sides by 1/2

4. A) x2 - y2 , x = 4 , y = 5 : given
42 - 52 : substitute x and y by the given values
=16 - 25 = -9

B) |4x - 2y| , x = -2 , y = 3 : given
|4(-2) - 2(3)| : substitute x and y by the given values
= |-14| = 14 : evaluate

C) 3x3 - 4y4 , x = -1 , y = -2 : given
3(-1)3 - 4(-2)4 : substitute x and y by the given values
= -3 - 64 = -67 : evaluate

5. A) x + 6 < 0 : given
x + 6 - 6 < -6 : subtract 6 from both sides
x < -6 : group like terms

B) x + 1 > 5 : given
x + 1 - 1 > 5 - 1 : subtract 1 from both sides
x > 4 : group like terms

C) 2(x - 2) < 12 : given
x - 2 < 6 : mutliply both sides by 1/2
x - 2 + 2 < 6 + 2 : add 2 to both sides
x < 8 : group like terms

6. A) (-1) a = 1 : definition: a is the reciprocal of -1
a = 1/-1 = -1 : solve for a; -1 is the reciprocal of -1

B) (0) b = 1 : definition: b is the reciprocal of 0
b = undefined : no value of b satisfies the above equation

C) (3/4) c = 1 : definition: c is the reciprocal of 3/4
c = 4/3 : solve for c; c = 4/3 is the reciprocal of 3/4

D) (2 5/7) d = 1 : definition: d is the reciprocal of 2 5/7.
(19/7) d = 1 : convert the mixed number 2 5/7 into a fraction.
d = 7/19 : : solve for d; d = 7/19 is the reciprocal of 2(5/7)

E) 0.02 d = 1 : definition: d is the reciprocal of 0.02.
d = 1/0.02 : solve for d; d = 50 is the reciprocal of 0.02

7. A) 3 3/4 + 6 1/7 : given
= (3 + 6) + (3/4 + 1/7) : put the whole parts together and the fractional parts together.
= 9 + (21/28 + 4/28) : add.
= 9 25/28

B) (1 3/5) × (3 1/3) - 2 1/2 : given
= (8/5) × (10/3) - 2 1/2 : convert mixed numbers to mupliply into fractions.
= 80/15 - 2 1/2 = 5 1/3 - 2 1/2 = 4 4/3 - 2 1/2: mupliply and write as mixed number if possible
= (4 - 2) + (4/3 - 1/2) : subtract
= 2 5/6

C) (5 2/3) ÷ (4 1/5) : given
= (17/3) ÷ (21/5) : convert mixed numbers to fractions.
= 85 / 63 : divide fractions
= 1 22/63 : write as mixed number

D) (3 4/7 - 1 1/2) ÷ (2 3/8 + 2 1/4) : given
= [(3 - 1) + (4/7 - 1/2)] ÷ [(2 + 2) + (3/8 + 1/4)]: evaluate numerator and denominator as fractions.
= (2 1/14) ÷ (4 5/8)
= (29/14) ÷ (37/8)
= 116/259

8. A) - 42 = - (4 × 4) = -16 : expand and calculate

B) (-2)3 = (-2)×(-2)×(-2) = -8 : expand and calculate

C) 10000 = 1 : definition: any nonzero number to the power zero gives 1

D) 5661 = 566

9. A) 0.02 = 1/50
B) 12% = 3/25
C) 0.5% = 1/200
D) 1.12 = 28/25

10. A) 1/5 = 0.2
B) 120% = 1.2
C) 0.2% = 0.002
D) 4 8/5 = 5.6

11. A) 3/10 = 30%
B) 1.4 = 140%
C) 123.45 = 12345%
D) 2 4/5 = 280%

12. A) 156312 , is divisible by 3
B) 176314 , is not divisible by 3

13. A) 3432 , is divisible by 4
B) 1257 , is not divisible by 4

14. A) 1233 , is not divisible by 6
B) 3432 , is divisible by 6

15. A) 2538 , is divisible by 9
B) 1451 , is not divisible by 9

16. Evaluate 8x + 7 given that x - 3 = 10.
x - 3 = 10: given equation
x = 10 + 3 = 13: solve the given equation.
8(13) + 7 = 111 substitute x by 3 in the given expression and evaluate.