Sure, let's go through the steps to solve each part of the question using the distributive property.
### Part (a)
We are given:
[tex]\[ -6(9 + x) = -54 + 48 \][/tex]
Step 1: Apply the distributive property on the left-hand side:
[tex]\[ -6 \cdot 9 + (-6 \cdot x) \][/tex]
[tex]\[ -54 - 6x \][/tex]
Step 2: Simplify the right-hand side:
[tex]\[ -54 + 48 \][/tex]
[tex]\[ -6 \][/tex]
So, the equation simplifies to:
[tex]\[ -54 - 6x = -6 \][/tex]
Step 3: Isolate the term involving [tex]\( x \)[/tex] by adding 54 to both sides:
[tex]\[ -54 - 6x + 54 = -6 + 54 \][/tex]
[tex]\[ -6x = 48 \][/tex]
Step 4: Solve for [tex]\( x \)[/tex] by dividing both sides by -6:
[tex]\[ x = \frac{48}{-6} \][/tex]
[tex]\[ x = -8 \][/tex]
So, the value of [tex]\( x \)[/tex] that makes the statement true is:
[tex]\[ x = -8 \][/tex]
### Part (b)
We are given:
[tex]\[ 6(x - 9) = 42 - 63 \][/tex]
Step 1: Simplify the right-hand side:
[tex]\[ 42 - 63 \][/tex]
[tex]\[ -21 \][/tex]
So, the equation simplifies to:
[tex]\[ 6(x - 9) = -21 \][/tex]
Step 2: Apply the distributive property on the left-hand side:
[tex]\[ 6x - 54 = -21 \][/tex]
Step 3: Isolate the term involving [tex]\( x \)[/tex] by adding 54 to both sides:
[tex]\[ 6x - 54 + 54 = -21 + 54 \][/tex]
[tex]\[ 6x = 33 \][/tex]
Step 4: Solve for [tex]\( x \)[/tex] by dividing both sides by 6:
[tex]\[ x = \frac{33}{6} \][/tex]
[tex]\[ x = \frac{11}{2} \][/tex]
[tex]\[ x = 5.5 \][/tex]
So, the value of [tex]\( x \)[/tex] that makes the statement true is:
[tex]\[ x = 5.5 \][/tex]
In summary, the values that make each statement true are:
- For (a), [tex]\( x = -8 \)[/tex]
- For (b), [tex]\( x = 5.5 \)[/tex]
