Let's solve the equation [tex]\( |8 + 2x| + 2x = 40 \)[/tex].
Since we have an absolute value in the equation, we need to consider two possible cases for the expression inside the absolute value.
Case 1: [tex]\( 8 + 2x \geq 0 \)[/tex]
In this case, the absolute value function can be removed without changing the sign:
[tex]\[ |8 + 2x| = 8 + 2x \][/tex]
Substitute back into the original equation:
[tex]\[ 8 + 2x + 2x = 40 \][/tex]
Combine like terms:
[tex]\[ 8 + 4x = 40 \][/tex]
Subtract 8 from both sides:
[tex]\[ 4x = 32 \][/tex]
Divide both sides by 4:
[tex]\[ x = 8 \][/tex]
Now we need to check if our solution [tex]\( x = 8 \)[/tex] fits the condition [tex]\( 8 + 2x \geq 0 \)[/tex]:
[tex]\[ 8 + 2(8) = 8 + 16 = 24 \geq 0 \][/tex]
Since the condition is satisfied, [tex]\( x = 8 \)[/tex] is a valid solution.
Case 2: [tex]\( 8 + 2x < 0 \)[/tex]
In this case, the absolute value function will change the sign of the expression inside:
[tex]\[ |8 + 2x| = -(8 + 2x) = -8 - 2x \][/tex]
Substitute back into the original equation:
[tex]\[ -8 - 2x + 2x = 40 \][/tex]
Since the [tex]\( -2x \)[/tex] and [tex]\( +2x \)[/tex] cancel each other out, we have:
[tex]\[ -8 = 40 \][/tex]
This is a contradiction because [tex]\(-8 \neq 40\)[/tex]. Therefore, there are no solutions from this case.
Conclusion:
The only solution to the equation [tex]\( |8 + 2x| + 2x = 40 \)[/tex] is:
[tex]\[ x = 8 \][/tex]
