Question

1. $5 + |9 - 3k| = 11$
2. $8 - 8|6r + 4| = 2$

Explanation:

Step1: Solve $5 + |9 - 3k| = 11$: Isolate absolute value

Subtract 5 from both sides:
$|9 - 3k| = 11 - 5$
$|9 - 3k| = 6$

Step2: Split into two linear equations

For absolute value equality $|A|=B$, $A=B$ or $A=-B$:
$9 - 3k = 6$ or $9 - 3k = -6$

Step3: Solve first linear equation

Subtract 9, then divide by -3:
$-3k = 6 - 9$
$-3k = -3$
$k = \frac{-3}{-3} = 1$

Step4: Solve second linear equation

Subtract 9, then divide by -3:
$-3k = -6 - 9$
$-3k = -15$
$k = \frac{-15}{-3} = 5$

Step5: Solve $8 - 8|6r + 4| = 2$: Isolate absolute value

Subtract 8, then divide by -8:
$-8|6r + 4| = 2 - 8$
$-8|6r + 4| = -6$
$|6r + 4| = \frac{-6}{-8} = \frac{3}{4}$

Step6: Split into two linear equations

For $|A|=B$, $A=B$ or $A=-B$:
$6r + 4 = \frac{3}{4}$ or $6r + 4 = -\frac{3}{4}$

Step7: Solve first linear equation

Subtract 4, then divide by 6:
$6r = \frac{3}{4} - 4 = \frac{3}{4} - \frac{16}{4} = -\frac{13}{4}$
$r = \frac{-\frac{13}{4}}{6} = -\frac{13}{24}$

Step8: Solve second linear equation

Subtract 4, then divide by 6:
$6r = -\frac{3}{4} - 4 = -\frac{3}{4} - \frac{16}{4} = -\frac{19}{4}$
$r = \frac{-\frac{19}{4}}{6} = -\frac{19}{24}$

Answer:

For $5 + |9 - 3k| = 11$: $k=1$ or $k=5$
For $8 - 8|6r + 4| = 2$: $r=-\frac{13}{24}$ or $r=-\frac{19}{24}$