solve the absolute value problem : 3x+|4x+5|=10
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign.
So, for example, the absolute value of 3 is 3, and the absolute value of
-3 is also 3.
In this case:
| 4 x + 5 | = 4 x + 5
OR
| 4 x + 5 | = - ( 4 x + 5 ) = - 4 x - 5
Equation:
3 x + | 4 x + 5 | = 10
has two solutions.
1.)
3 x + 4 x + 5 = 10
7 x = 10 - 5
7 x = 5 Divide both sides with 7
x = 5 / 7
2.)
3 x - ( 4 x + 5 ) = 10
3 x - 4 x - 5 = 10
- x = 10 + 5
- x = 15 Multiply both sides with - 1
x = - 15
| 4 x + 5 | = 4 x + 5
| - ( 4 x + 5 ) | = | - 4 x - 5 | = | 4 x + 5 |
More direct way:
3x+|4x+5|=10
|4x+5| = 10-3x
4x+5 = 10-3x OR -4x-5 = 10-3x
7x = 5 OR -x = 15
x = 5/7 OR x = -15
To solve the absolute value problem 3x + |4x + 5| = 10, we can break it down into two cases based on the expression inside the absolute value:
Case 1: (4x + 5) ≥ 0
In this case, the absolute value |4x + 5| is equal to (4x + 5). Therefore, the equation becomes:
3x + (4x + 5) = 10
Simplify the equation:
7x + 5 = 10
7x = 5
x = 5/7
Case 2: (4x + 5) < 0
In this case, the absolute value |4x + 5| is equal to -(4x + 5), which means we need to change its sign. Therefore, the equation becomes:
3x - (4x + 5) = 10
Simplify the equation:
3x - 4x - 5 = 10
-x - 5 = 10
-x = 15
x = -15
So the two solutions to the absolute value problem are x = 5/7 and x = -15.