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Algebra
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Equations and Inequalities
Identifying Properties When Solving Equations
Independent Practice
 
1. The following pairs of equations are equivalent. Describe the operation that occurred in the second equation.
 
 
 
 
 
 
2. Complete the following table with the properties used to solve 4(x+3)=20.
 
Statement Proof
4 (x+3) = 20 Given
4x + 12 = 20  
4x = 8  
x = 2  

3. Complete the following table with the mathematical statements that correspond to the proofs used to solve  

Statement Proof
  Given
  Multiplication Property of Equality
  Distributive Property
  Addition Property of Equality
  Division Property of Equality
 
4. Consider the equations 5x + 10 = 30 and 5 (x+10) = 30.
Do they have the same solution? Why or why not?
 
5. Consider the equations 3x + 2 = 14 and 2+3x = 14.
Do they have the same solution? Why or why not?
 
6. Consider the equation 3 (x+2)+3x = 36
Without solving. Name all the properties that would be used to solve the equation.
 
7. Consider the equation  
Part A: Write an equivalent equation using the multiplication property of equality.
Part B: What properties will you use next to solve the equation?
8. Solve the following equation. Justify each step.
0.2x + 3.1 - 2.1x = 0.3 (x-5) + 0.2
 

 

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Related questions

Statements Proof
  Given
  Multiplication Property of Equality
  Distributive Property
  Addition Property of Equality
  Division Property of Equality
 
4. Consider the equations 5 x+10=30 and 5(x+10)=30.
    Do they have the same solution? Why or why not?
 
5. Consider the equations 3 x+2=14 and 2+3x=14.
    Do they have the same solution? Why or why not?
 
6. Consider the equation 3(x+2)+3 x=36
    Without solving. name all the properties that would be used to solve the equation.
Equations and Inequalities
Equations: True or False?
Independent Practice
1. Consider the statement: 4 - 3 + 5 = -6 + 8 + 4. This is a mathematically correct sentence.
Is the sentence true or false? Explain how you know.
 
2. Determine if the sentence is true. Select all that apply.
    ⬤   
    ⬤   
    ⬤   
    ⬤   
    ⬤   
 
3. Determine whether the following number sentences are TRUE or FALSE. Justify your answer.
 
                                        
  
 
                                                       
 
 
4. For the equation, x-7=24, can a value be substituted for x to make that a true number sentence? How many values could be substituted for x and have a true number sentence?
 
5. Consider x+4=x+8 . What values could be substituted for x to make this a true number sentence? Explain how you know.
 
6. Determine what value(s) for the variable would make each algebraic equation a true number sentence.
is true for                      _______________.
is true for         _______________.
  is true for         _______________.
 
7. Which of the following has the correct solution given for x? Check all that apply.
 
  ⬤    
  ⬤   
  ⬤    
  ⬤    
  ⬤    
 
 

Q18. A size 6 dress in Country C is size 52 in Country D. A function that converts dress sizes in Country C to those in Country D is f(x) = 2(x + 20). Find a formula for the inverse of the function described.
a. f -1(x) = (x - 20)/2
b. f -1(x) = x - 20
c. f -1(x) = (x/2) - 20
d. f -1(x) = (x/2) + 20

Q19. If f(x) = 4x and g(x) = 12, find the point of intersection of the graphs of f and g by solving f(x) = g(x). Give an exact answer.
a. (log 4 12, 12)
b. (log 4 12, 4)
c. (log 4 12, 0)
d. (12, 12)

Q20. Find functions f and g so that f ∘ g = H(x) = (5 - 2x3)2.
a. f(x) = 5 - 2x3 ; g(x) = x2
b. f(x) = (5 - 2x)3 ; g(x) = x2
c. f(x) = x2 ; g(x) = 5 - 2x3
d. f(x) = x3 ; g(x) = (5 - 2x)2

Q29. Solve the system of equations using Cramer's Rule if it is applicable.


a. x = 9, y = 2, z = 5; (9, 2, 5)
b. x = 8, y = 4, z = 5; (8, 4, 5)
c. x = 4, y = 5, z = 4; (4, 5, 4)
d. x = 8, y = -4, z = -5; (8, -4, -5)

Q30. Solve the system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.


a. x = 2, y = 3, z = -7; (2, 3, -7)
b. x = 1, y = 3, z = -4; (1, 3, -4)
c. x = 5, y = 13/2, z = -3; (5, 13/2, -3)
d. x = 1, y = -3, z = -16; (1, -3, -16)

Q33. Write the partial fraction decomposition of the rational expression (4x3 + 4x2)/(x2 + 5)2.
a. (4x - 4)/(x2 + 5) + (-20x + 20)/(x2 + 5)2
b. (4x + 4)/(x2 + 5) + (-20x - 20)/(x2 + 5)2
c. (4x + 4)/(x2 + 5) + (20x - 20)/(x2 + 5)2
d. (4x + 4)/(x2 + 5) + (20x + 20)/(x2 + 5)2

Q34. Solve the linear programming problem. Minimize z = 11x + 6y + 7 subject to: x = 0, y = 0, x + y = 1.
a. minimum: 7
b. minimum: 18
c. minimum: 24
d. minimum: 13

Q35. Solve the system of equations by elimination.


a. x = 10, y = 0; (10, 0)
b. x = 10, y = 10; (10, 10)
c. x = 0, y = 10; (0, 10)
d. x = 0, y = 0; (0, 0)

Q36. Perform the row operations on the given augmented matrix.


a.
b.
c.
d.

Q37. Solve the system of equations using Cramer's Rule if it is applicable.


a. x = -4, y = 3; (-4, 3)
b. x = -3, y = -4; (-3, -4)
c. x = 4, y = 3; (4, 3)
d. x = 3, y = 4; (3, 4)

Q38. Write the partial fraction decomposition of the rational expression x/(x2 - 9x + 20).
a. -4/(x - 4) + 5/(x - 5)
b. -5/(x - 4) + 4/(x - 5)
c. 4/(x - 4) + -5/(x - 5)
d. -4/(x - 4) + -5/(x - 5)

Q39. Solve the system of equations.


a. x = -4, y = 4, z = -3; (-4, 4, -3)
b. x = -3, y = 4, z = -4; (-3, 4, -4)
c. x = -4, y = -3, z = 4; (-4, -3, 4)
d. inconsistent

Q40. Find the value of the determinant.


a. 224
b. 24
c. -72
d. 72

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