To solve the equation [tex]\( |6u + 3| = 9 \)[/tex], we need to consider the definition of absolute value. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, [tex]\( |6u + 3| = 9 \)[/tex] means that [tex]\( 6u + 3 \)[/tex] can be either 9 or -9.
We can then set up two separate equations to solve for [tex]\( u \)[/tex]:
Case 1:
[tex]\[ 6u + 3 = 9 \][/tex]
Subtract 3 from both sides:
[tex]\[ 6u = 6 \][/tex]
Divide both sides by 6:
[tex]\[ u = 1 \][/tex]
Case 2:
[tex]\[ 6u + 3 = -9 \][/tex]
Subtract 3 from both sides:
[tex]\[ 6u = -12 \][/tex]
Divide both sides by 6:
[tex]\[ u = -2 \][/tex]
Therefore, the solutions for [tex]\( u \)[/tex] are [tex]\( u = 1 \)[/tex] and [tex]\( u = -2 \)[/tex].
So, the solutions are:
[tex]\[ u = 1, -2 \][/tex]
