Why does |7 + 3x| = x -1 have no solution? I could plug in, for example 5, and the answer would be positive.
Thank you =)
you might plug in x = 5, but x=5 does not satisfy the equation, so it is not a solution. So what if both sides are positive.
We know by definition that x-1 > 0 or
x>= 1
7+3x = x-1 or -7-3x = x-1
2x = -8 or -4x = 6
x = -4 or x = -3/2
subbing either one of these does not satisfy the equation, thus no solution
or
we established that x >= 1
and neither of the solutions satisfy this original condition, so there is no solution
Thanks. That cleared up a lot of thinfs
To understand why the equation |7 + 3x| = x - 1 has no solution, we need to examine the properties of absolute value and the equation itself.
The equation |7 + 3x| = x - 1 involves an absolute value expression. The absolute value of a number (denoted by |x|) is always non-negative, meaning it is either zero or positive. Therefore, |7 + 3x| can never be negative.
Now, let's consider the equation |7 + 3x| = x - 1. In order to solve this equation, we need to consider two cases: when the expression inside the absolute value is positive and when it is zero.
Case 1: When 7 + 3x is positive (greater than 0).
For this case, we can rewrite the equation as (7 + 3x) = x - 1. By simplifying, we get 3x = x - 8, which leads to 2x = -8, and finally, x = -4. However, we need to check if this solution satisfies the initial condition. Plugging in x = -4 into the original equation gives us |7 + 3(-4)| = -4 - 1, which simplifies to |-5| = -5. Since the absolute value of -5 is 5 (a positive value), this solution does not work in this case.
Case 2: When 7 + 3x is zero.
For this case, we can rewrite the equation as |0| = x - 1. The absolute value of zero is zero, and subtracting 1 from zero gives us -1. So, one potential solution is x = -1.
Now, let's put this solution into the original equation: |7 + 3(-1)| = -1 - 1, which simplifies to |-4| = -2. However, the absolute value of -4 is 4 (a positive value), not -4. Therefore, x = -1 is not a solution either.
Since both cases produced no valid solution, we can conclude that the equation |7 + 3x| = x - 1 has no solutions.