What is the solution set for the equation $| 4 a + 6 | - 4 a = 10$ $|4a + 6| − 4a = 10$ ?

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Answer 1 · 2015-10-13T17:02:03

$a = - 2$ $a = -2$

The first thing to do here is isolate the modulus on onse side of the equation by adding $4 a$ $4a$ to both sides

$|4a + 6| - color(red)(cancel(color(black)(4a))) + color(red)(cancel(color(black)(4a))) = 10 + 4a$

$|4a + 6| = 10 + 4a$

Now, by definition, the absolute value of a real number will only return positive values, regardless of the sign of said number.

This means that the first condition that any value of $a$ must satisfy in order to be a valid solution will be

$10 + 4a >= 0$

$4a >= -10 implies a >= -5/2$

Keep this in mind. Now, since the absolute value of a number returns a positive value, you can have two possibilities

$4a + 6 < 0 implies |4a + 6| = -(4a+6)$

In this case, the equation becomes

$-(4a + 6) = 10 + 4a$

$-4a - 6 = 10 + 4a$

$8a = - 16 implies a = ((-16))/8 = -2$

$(4a + 6) >=0 implies |4a + 6| = 4a + 6$

This time, the equation becomes

$color(red)(cancel(color(black)(4a))) + 6 = 10 + color(red)(cancel(color(black)(4a)))$

$6 != 10 implies a in O/$

Therefore, the only valid solution will be $a = -2$ . Notice that it satisfies the initial condition $a >= -5/2$ .

Do a quick check to make sure that the calculations are correct

$|4 * (-2) + 6| - 4 * (-2) = 10$

$|-2| +8 = 10$

$2 + 8 = 10 color(white)(x)color(green)(sqrt())$

What is the solution set for the equation |4a + 6| − 4a = 10|4a+6|−4a=10|4a + 6| − 4a = 10 ?