solve for x

Question

Let
g(x) = 4x − 2 + |x + 3|.
Find all values of a which satisfy the equation
g(a) = 3a + 5.

I know you replace the x's with a but i cant seem to get the correct answers.

Answer 1

Remember that whenever you solve an absolute value equation, you have to isolate the absolute value on one side of the equation.  Then the contents of the absolute value either equals what is on the other side OR

the contents of the absolute value equals -(what is on the other side).

 

If g(x) = 4x − 2 + |x + 3|, and you want to find all values of a which satisfy the equation g(a) = 3a + 5,

first isolate the |x + 3|:

 

4a - 2 + |a + 3| = 3a + 5  Subtract 4a from both sides, add 2 to both sides and you get

 

either 1) |a + 3| = 7-a

 

OR     2) |a + 3| = -(7-a)

 

Now solve both of these cases for a.

 

Lynne R.

Answer 2

Jon,

 

First simplify the g(x) to remove the absolute value sign by considering both the + and - cases.  Then solve the simplified equation for a by applying your given g(a) constraint.

 

g(x) = 4x - 2  + (x + 3)  =  5x + 1  , the positive case for absolute value

g(x) = 4x - 2  -  (x + 3)  =  3x  -5   , the negative case for absolute value

 

Using the first eq:   g(a) = 5a + 1  =  3a + 5  (the constraint)

So  2a = 4   and  a = 2

 

Using the other eq: g(a) = 3a - 5   =  3a + 5 (the constraint),  which has no solution for a

 

In summary, the sought after solution is:

g(x) = 5x + 1  and x = a = 2  for the constraint to be true since g(a) = 5 (2) + 1 = 3(2) + 5 = 11

 

 

Answer 3

If you look at the graph of |x|, its a line with slope 1 if x is positive or a line of slope of -1 if x is negative. 

 

 

|x + 3| is the same graph just shifted 3 units to the left.

 

 

Notice that |x + 3| = x + 3 when x > -3  or  |x + 3| = -x - 3 when x < -3. 

 

 

Use these separately to solve for x.  Good luck.

 

 

 

By the way, when x = -3, |x + 3| = 0.