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If |x + 1| = 2|x - 1|, what is the sum of the roots?
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17 Jul 2017, 08:02

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Amby02 wrote:

If |x + 1| = 2|x - 1|, what is the sum of the roots?

A. 4 B. 6 C. 8 D. 20/3 E. 10/3

The only way the two sides will be equal is if quantities inside the absolute value brackets are 1) equal or 2) equal but with opposite signs. |x| = y or -y*

1. Remove the brackets

2. LHS = RHS or LHS = -RHS*

3. Set up the two equations

CASE 1: x + 1 = 2(x - 1) OR

CASE 2: x + 1 = -[2(x -1)]

4. Solve

CASE 1: x + 1 = 2x - 2 3 = x ... x = 3

CASE 2: x + 1 = -[2(x -1)] x + 1 = -(2x - 2) x + 1 = -2x + 2 3x = 1 x = \(\frac{1}{3}\)

5. Check x = 3 and x = \(\frac{1}{3}\) When removing absolute value brackets, I always check to see whether or not the roots satisfy the original equation. Both work.

6. Sum of roots is 3 + \(\frac{1}{3}\) = \(\frac{10}{3}\)

Answer D

*(and |y| = x or -x). You can reverse RHS and LHS, where RHS = LHS or RHS = - LHS. pushpitkc shows the four possibilities. The latter two are identical to the first two.
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Re: If |x + 1| = 2|x - 1|, what is the sum of the roots?
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18 Jul 2017, 06:55

Amby02 wrote:

If |x + 1| = 2|x - 1|, what is the sum of the roots?

A. 4 B. 6 C. 8 D. 20/3 E. 10/3

whenever there is absolute sign on both sides,,square it off... gives (x+1)^2 = 4 (x-1)^2 x^2 + 2x + 1 = 4x^2 + 4 - 8x 3x^2 - 10x + 3 = 0 sum of both the roots = -b/a = 10/3

Which of these is the sum of the solutions of |x + 1| = 2|x - 1|?
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Updated on: 20 Jan 2018, 10:40

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Top Contributor

fraserlawson wrote:

Which of these is the sum of the solutions of |x+1|=2|x-1|?

A) 4 B) 6 C) 8 D) 20/3 E) 10/3

There are 3 steps to solving equations involving ABSOLUTE VALUE: 1. Apply the rule that says: If |x| = k, then x = k and/or x = -k 2. Solve the resulting equations 3. Plug solutions into original equation to check for extraneous roots

Given: |x+1|=2|x-1|

So, EITHER x+1=2(x-1) OR x+1=-[2(x-1)]

x+1=2(x-1) Expand: x + 1 = 2x - 2 Solve: x = 3

x+1=-[2(x-1)] Expand: x + 1 = -2x + 2 Solve: x = 1/3

When we plug the solutions into the original equation, we find that there are no extraneous roots

So, the SUM of the solutions = 3 + 1/3 = 3 1/3 = 10/3

Answer: E

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Originally posted by GMATPrepNow on 20 Jan 2018, 08:58.
Last edited by GMATPrepNow on 20 Jan 2018, 10:40, edited 1 time in total.

Re: Which of these is the sum of the solutions of |x + 1| = 2|x - 1|?
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20 Jan 2018, 10:03

GMATPrepNow wrote:

There are 3 steps to solving equations involving ABSOLUTE VALUE: 1. Apply the rule that says: If |x| = k, then x = k and/or x = -k 2. Solve the resulting equations 3. Plug solutions into original equation to check for extraneous roots

Given: |x+1|=2|x-1|

So, EITHER x+1=2(x-1) OR EITHER x+1=-[2(x-1)]

x+1=2(x-1) Expand: x + 1 = 2x - 2 Solve: x = 3

x+1=-[2(x-1)] Expand: x + 1 = -2x + 2 Solve: x = 1/3

When we plug the solutions into the original equation, we find that there are no extraneous roots

So, the SUM of the solutions = 3 + 1/3 = 3 1/3 = 10/3

Answer: E

Awesome way to solve the problem, and way faster than what I did to brute-force it. Thank you!

One question, though. How is this approach different from how one would solve |x+1|=2(x-1)? Or x+1 = 2|x-1|?

I ask because 2|x-1|-(x+1)=0 has roots of 1/3 and 3, whereas 2(x-1)-|x+1|=0 only has the root at 3 (unless I've used Wolfram wrong). Is there something about knowing which one of the two absolute value elements to pick?

Re: Which of these is the sum of the solutions of |x + 1| = 2|x - 1|?
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20 Jan 2018, 10:40

Top Contributor

fraserlawson wrote:

GMATPrepNow wrote:

There are 3 steps to solving equations involving ABSOLUTE VALUE: 1. Apply the rule that says: If |x| = k, then x = k and/or x = -k 2. Solve the resulting equations 3. Plug solutions into original equation to check for extraneous roots

Given: |x+1|=2|x-1|

So, EITHER x+1=2(x-1) OR x+1=-[2(x-1)]

x+1=2(x-1) Expand: x + 1 = 2x - 2 Solve: x = 3

x+1=-[2(x-1)] Expand: x + 1 = -2x + 2 Solve: x = 1/3

When we plug the solutions into the original equation, we find that there are no extraneous roots

So, the SUM of the solutions = 3 + 1/3 = 3 1/3 = 10/3

Answer: E

Awesome way to solve the problem, and way faster than what I did to brute-force it. Thank you!

One question, though. How is this approach different from how one would solve |x+1|=2(x-1)? Or x+1 = 2|x-1|?

I ask because 2|x-1|-(x+1)=0 has roots of 1/3 and 3, whereas 2(x-1)-|x+1|=0 only has the root at 3 (unless I've used Wolfram wrong). Is there something about knowing which one of the two absolute value elements to pick?

Good question. When there are absolute values on BOTH sides, it doesn't matter which one you choose to make negative.

For example, with |x+1|=2|x-1|, we could have also gone with: EITHER x+1=2(x-1) OR -(x+1)=2(x-1) The second equation still results in x = 1/3

Cheers, Brent
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Re: Which of these is the sum of the solutions of |x + 1| = 2|x - 1|? &nbs
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