To solve the equation [tex]\( |2x - 19| = 4x + 1 \)[/tex], we need to consider the definition of the absolute value function. The absolute value expression [tex]\( |2x - 19| \)[/tex] can be broken down into two separate cases:
Case 1:
[tex]\[ 2x - 19 = 4x + 1 \][/tex]
Case 2:
[tex]\[ 2x - 19 = -(4x + 1) \][/tex]
Let's solve each case separately.
### Case 1: [tex]\( 2x - 19 = 4x + 1 \)[/tex]
First, we isolate [tex]\(x\)[/tex]:
[tex]\[ 2x - 19 = 4x + 1 \][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[ -19 = 2x + 1 \][/tex]
Subtract 1 from both sides:
[tex]\[ -20 = 2x \][/tex]
Divide both sides by 2:
[tex]\[ x = -10 \][/tex]
We need to check if this solution is valid by substituting [tex]\(x = -10\)[/tex] back into the original equation:
[tex]\[ |2(-10) - 19| = 4(-10) + 1 \][/tex]
[tex]\[ |-20 - 19| = -40 + 1 \][/tex]
[tex]\[ |-39| = -39 \][/tex]
[tex]\[ 39 \neq -39 \][/tex]
Since the left-hand side does not equal the right-hand side, [tex]\( x = -10 \)[/tex] is not a valid solution.
### Case 2: [tex]\( 2x - 19 = -(4x + 1) \)[/tex]
First, distribute the negative sign on the right-hand side:
[tex]\[ 2x - 19 = -4x - 1 \][/tex]
Next, we isolate [tex]\(x\)[/tex]:
Add [tex]\(4x\)[/tex] to both sides:
[tex]\[ 6x - 19 = -1 \][/tex]
Add 19 to both sides:
[tex]\[ 6x = 18 \][/tex]
Divide both sides by 6:
[tex]\[ x = 3 \][/tex]
We need to check if this solution is valid by substituting [tex]\(x = 3\)[/tex] back into the original equation:
[tex]\[ |2(3) - 19| = 4(3) + 1 \][/tex]
[tex]\[ |6 - 19| = 12 + 1 \][/tex]
[tex]\[ |-13| = 13 \][/tex]
[tex]\[ 13 = 13 \][/tex]
Since the left-hand side equals the right-hand side, [tex]\( x = 3 \)[/tex] is a valid solution.
### Conclusion
The only solution to the equation [tex]\( |2x - 19| = 4x + 1 \)[/tex] is:
[tex]\[ x = 3 \][/tex]
