Quandaries and Queries




Without the use of trial and error, is there a way of solving for x where |x + 3| = -15x? I know that x = -0.1875, but how do I find whether x + 3 is positive or not if I did not have this information (In other words, how would I solve whether the x-value was -3 or less, or if it was more than -3)?



Hi Nathan,

Usually there is not a way to tell if x+3 is positive or negative. There are however at least two ways to solve this problem.

First, you could square both sides to get rid of the absolute values but then you would get two solutions and need to figure out which of the two is the right one.The squaring gives

(|x+3|)2 = (-15x)2

which is the quadratic

x2 + 6x + 9 = 225x2

A second technique is to treat the two cases, x+3 is positive or negative, separately. If x+3 is positive then |x+3|=x+3 and hence the equation is

x + 3 = -15x

If x+3 is negative then |x+3|=-(x+3) and the equation is

-(x+3) = -15x

Solve each of these equations to get again two solutions and check to see which is correct.

Claude and Penny


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