49. Reasoning Determine whether each statement is always, sometimes, or never true.
    1. An equation of the form ax + 1 = ax has no solution.
    2. An equation in one variable has at least one solution.
    3. An equation of the form x a = x b x over eh , equals , x over b  has infinitely many solutions.

C Challenge

Open-Ended Write an equation with a variable on both sides such that you get each solution.

50. x = 5
51. x = 0
52. x can be any number.
53. No values of x are solutions.
54. x is a negative number.
55. x is a fraction.
56. Suppose you have three consecutive integers. The greatest of the three integers is twice as great as the sum of the first two. What are the integers?

Standardized Test Prep

SAT/ACT

57. What is the solution of − 2 ( 3 x − 4 ) = − 2 x + 2 ? negative 2 open 3 x minus 4 close equals negative 2 x plus 2 question mark
    1. − 2 3 negative , 2 thirds
    2. 3 2 3 halves
    3. 2
    4. 24
58. Two times a number plus three equals one half of the number plus 12. What is the number?
    6. 3.6
    7. 6
    8. 8
    9. 10
59. Josie's goal is to run 30 mi each week. This week she has already run the distances shown in the table. She wants to have one day of rest and to spread out the remaining miles evenly over the rest of the week. Which equation can she use to find how many miles m per day she must run?

    Miles per Day

    M	T	W	T	F	S	S
    4	4.5	3.5	_____	_____	_____	_____

    1. 4 + 4.5 + 3.5 + 3m = 30
    2. 4 + 4.5 + 3.5 + 4m = 30
    3. 30 − ( 4 + 4.5 + 3.5 ) = m 30 minus open 4 plus 4.5 plus 3.5 close equals m
    4. 4 + 4.5 + 3.5 + m = 30

Mixed Review

See Lesson 2-3.

Solve each equation.

60. − 2 a + 5 a − 4 = 11 negative 2 eh plus 5 eh minus 4 equals 11
61. 6 = − 3 ( x + 4 ) 6 equals negative 3 open x plus 4 close
62. 3 ( c + 1 3 ) = 4 3 . open . c plus , 1 third . close . equals 4
63. A carpenter is filling in an open entranceway with a door and two side panels of the same width. The entranceway is 3 m wide. The door will be 1.2 m wide. How wide should the carpenter make the panels on either side of the door so that the two panels and the door will fill the entranceway exactly?

    See Lesson 2-2.

Get Ready! To prepare for Lesson 2-5, do Exercises 64–66.

See Lesson 1-2.

Evaluate each expression for the given values of the variables.

64. n + 2m; m = 12, n = − 2 n equals negative 2
65. 3 b ÷ c ; 3 b divides c semicolon  b = 12, c = 4
66. x y 2 ; x , y squared , semicolon  x = 2.8, y = 2

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