Consider the sequence of steps to solve the equation: 3(x − 4) + 5x = 9x − 36

Given ⇒ 3(x − 4) + 5x = 9x − 36
Step 1 ⇒ 3x − 12 + 5x = 9x − 36
Step 2 ⇒ 3x + 5x − 12 = 9x − 36
Step 3 ⇒ 8x − 12 = 9x − 36
Step 4 ⇒ 8x − 8x − 12 = 9x − 8x − 36
Step 5 ⇒ 0 − 12 = x − 36
Step 6 ⇒ −12 = x − 36
Step 7 ⇒ −12 + 36 = x − 36 + 36
Step 8 ⇒ 24 = x + 0
Step 9 ⇒ 24 = x

Which property yields Step 1?

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}

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step-by-step explanation:

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