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Math Help - What am I doing wrong?

  1. #1
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    What am I doing wrong?

    |2x+1|<|x+3| Solve for x.

    So the correct answer is -4/3<x<2 based on my deductions. But my problem is my solution generates more answers.

    +-(2x+1)<+-(x+3) =>

    1) 2x+1<x+3 => x<2
    2) 2x+1<-(x+3) => x<-4/3
    3) -(2x+1)<(x+3) =>x>-4/3
    4) -(2x+1)<-(x+3) => x>2


    What am I doing wrong in the solution to get 2), 4) as wrong answers. I know I am probably having a brain fart and missing something obvious.

    Thanks in advance for your help.
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: What am I doing wrong?

    I would write:

    \sqrt{(2x+1)^2}<\sqrt{(x+3)^2}

    Square both sides, and expand:

    4x^2+4x+1<x^2+6x+9

    3x^2-2x-8<0

    (3x+4)(x-2)<0

    Can you proceed from here?
    Thanks from aroe
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  3. #3
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    Re: What am I doing wrong?

    You need to consider three cases.

    When \displaystyle \begin{align*} x < -3 \end{align*} we have \displaystyle \begin{align*} |x + 3| = -(x + 3) \end{align*} and \displaystyle \begin{align*} |2x + 1| = -(2x + 1) \end{align*} .

    When \displaystyle \begin{align*} -3 \leq x < -\frac{1}{2} \end{align*} we have \displaystyle \begin{align*} |x + 3| = x + 3 \end{align*} and \displaystyle \begin{align*} |2x + 1| = -(2x + 1) \end{align*} .

    When \displaystyle \begin{align*} x \geq -\frac{1}{2} \end{align*} we have \displaystyle \begin{align*} |x + 3| = x + 3 \end{align*} and \displaystyle \begin{align*} |2x + 1| = 2x + 1 \end{align*} .

    So to solve \displaystyle \begin{align*} |2x + 1| < |x + 3| \end{align*} , when \displaystyle \begin{align*} x < -3 \end{align*} we have

    \displaystyle \begin{align*} -(2x + 1) &< -(x + 3) \\ 2x + 1 &> x + 3 \\ x + 1 &> 3 \\ x &> 2 \end{align*} .

    Since we don't have any values of x that are both < -3 and > 2, this is impossible.

    When \displaystyle \begin{align*} -3 \leq x < -\frac{1}{2} \end{align*} we have

    \displaystyle \begin{align*} -(2x + 1) &< x + 3 \\ 2x + 1 &> -x - 3 \\ 3x + 1 &> -3 \\ 3x &> -4 \\ x &> -\frac{4}{3} \end{align*}

    so that means \displaystyle \begin{align*} -\frac{4}{3} < x < -\frac{1}{2} \end{align*} is acceptable.

    When \displaystyle \begin{align*} x \geq -\frac{1}{2} \end{align*} we have

    \displaystyle \begin{align*} 2x + 1 &< x + 3 \\ x + 1 &< 3 \\ x &< 2 \end{align*}

    so that means \displaystyle \begin{align*} -\frac{1}{2} \leq x < 2 \end{align*} is also accetable.


    So that means our total solution is \displaystyle \begin{align*} -\frac{4}{3} < x < 2 \end{align*} .
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  4. #4
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    Re: What am I doing wrong?

    whenever i see |anything|, i think that this means:

    |a| = a, when a ≥ 0
    |a| = -a, when a < 0.

    now from appearances, it looks like we have 4 cases to investigate:

    1) 2x+1, x+3 ≥ 0
    2) 2x+1 ≥ 0, x+3 < 0
    3) 2x+1 < 0, x+3 > 0
    4) 2x+1,x+3 < 0

    let's look at each of these in turn.

    in case (1), we have |2x+1| < |x+3| becomes 2x+1 < x+3, which then can be manipulated algebraically:

    x+1 < 3
    x < 2

    but this is not quite all. if 2x+1 ≥ 0, this means that 2x ≥ -1, so x ≥ -1/2. since we need x+3 ≥ 0 as well, we need to simultaneously require x ≥ -3.

    the more restrictive of the two requirements is x ≥ -1/2. so if case 1 is the case, then x ≥ -1/2 > -3, in which case the inequality |2x+1| < |x+3| tells us x < 2, so we have:

    -1/2 ≤ x < 2.

    now we turn to case (2). here, 2x+1 ≥ 0 means that x ≥ -1/2, as before. but x+3 < 0, means x < -3. but x cannot be both ≥ -1/2 AND < -3 at the same time. so this case never occurs.

    on to case (3). here, 2x+1 < 0, means that x < -1/2. x+3 ≥ 0 means that x ≥ -3. so for case (3) to apply, we need -3 ≤ x < -1/2. but we still have the original inequality to satisfy, as well. for case (3), it is:

    -2x-1 < x+3
    -3x-1 < 3
    -3x < 4
    x > -4/3 (we have to "change direction" when we multiply by -1/3).

    since -4/3 is in-between -3 and -1/2, requiring that BOTH x > -4/3 AND -3 ≤ x < -1/2 translates into: -4/3 < x < -1/2.

    finally, case (4). here 2x+1 < 0 means x < -1/2, and x+3 < 0, means x < -3. the more restrictive condition is x < -3. in this case, our inequality is:

    -2x-1 < -x-3
    -x-1 < -3
    -x < -2
    x > 2.

    now since x cannot be both > 2 and < -3, in case (4) we have no x that qualify.

    now the choice between the 4 cases (only 2 of which are possible) is an "OR" choice. thus we have, as our final solution:

    -4/3 < x < -1/2 OR -1/2 ≤ x < 2 since the "endpoints match", we can combine these into the SINGLE condition:

    -4/3 < x < 2.

    here is how it works "in reverse":

    when x < -3, then 2x+1 < -5 < 0, and x+3 < 0, so condition (4) holds. but the only x that satisfy -2x-1 < -x-3 are x > 2, and no x < -3 is > 2.

    so there are no x's that work < -3.

    when -3 ≤ x < -1/2, then 2x+1 < 0, and x+3 ≥ 0, so condition (3) holds. but for -2x-1 < x+3, we have to have x > -4/3, so the x's that work are in:

    [-3,-1/2) ∩ (-4/3,∞) = (-4/3,-1/2).

    when -1/2 ≤ x, then 2x+1 ≥ 0 and x+3 ≥ 5/2 > 0, so condition (1) holds, in which case 2x+1 < x+3 means that x < 2. so that x's that work are in:

    [-1/2,∞) ∩ (-∞,2) = [-1/2,2).

    so our final answer is (-4/3,-1/2) U [-1/2,2) = (-4/3,2).

    **********

    let's compare this with MarkFL2's approach:

    suppose (3x + 4)(x - 2) < 0. when can this happen? in just two ways:

    A) 3x + 4 < 0 and x - 2 > 0

    B) 3x + 4 > 0 and x - 2 < 0

    (A) first: 3x + 4 < 0 means 3x < -4, and so x < -4/3. x - 2 > 0 means x > 2. it is impossible for both of these to be true.

    (B) next: 3x + 4 > 0 means x > -4/3, and x - 2 < 0 means x < 2. this, we can do, so we have: -4/3 < x < 2.

    much less work.
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