To solve the equation [tex]\( |3x| + 7x + 1 = 0 \)[/tex], we need to recognize that the absolute value function [tex]\( |3x| \)[/tex] can be split into two separate cases based on the sign of [tex]\( x \)[/tex].
### Case 1: [tex]\( 3x \geq 0 \)[/tex]
When [tex]\( 3x \geq 0 \)[/tex], the absolute value function [tex]\( |3x| \)[/tex] simplifies to [tex]\( 3x \)[/tex]. So the equation becomes:
[tex]\[ 3x + 7x + 1 = 0 \][/tex]
This simplifies to:
[tex]\[ 10x + 1 = 0 \][/tex]
### Case 2: [tex]\( 3x < 0 \)[/tex]
When [tex]\( 3x < 0 \)[/tex], the absolute value function [tex]\( |3x| \)[/tex] simplifies to [tex]\( -3x \)[/tex]. So the equation becomes:
[tex]\[ -3x + 7x + 1 = 0 \][/tex]
This simplifies to:
[tex]\[ 4x + 1 = 0 \][/tex]
Therefore, the system of equations based on the two cases is:
1. [tex]\( 10x + 1 = 0 \)[/tex]
2. [tex]\( 4x + 1 = 0 \)[/tex]
Next, we solve these two equations individually to find the possible values of [tex]\( x \)[/tex].
### Solving the First Equation
[tex]\[ 10x + 1 = 0 \][/tex]
Subtract 1 from both sides:
[tex]\[ 10x = -1 \][/tex]
Divide both sides by 10:
[tex]\[ x = -\frac{1}{10} \][/tex]
### Solving the Second Equation
[tex]\[ 4x + 1 = 0 \][/tex]
Subtract 1 from both sides:
[tex]\[ 4x = -1 \][/tex]
Divide both sides by 4:
[tex]\[ x = -\frac{1}{4} \][/tex]
### Conclusion
The solutions to the equation [tex]\( |3x| + 7x + 1 = 0 \)[/tex] are:
[tex]\[ x = -\frac{1}{10} \][/tex]
and
[tex]\[ x = -\frac{1}{4} \][/tex]
Thus, the system of equations that could be used to solve [tex]\( |3x| + 7x + 1 = 0 \)[/tex] is:
1. [tex]\( 10x + 1 = 0 \)[/tex]
2. [tex]\( 4x + 1 = 0 \)[/tex]
And the solutions are [tex]\( x = -\frac{1}{10} \)[/tex] and [tex]\( x = -\frac{1}{4} \)[/tex].
