To determine if [tex]\( x = 10 \)[/tex] is a solution to the inequality [tex]\( -6(-4 + x) < 120 + 4x \)[/tex], let's evaluate both sides of the inequality step-by-step for [tex]\( x = 10 \)[/tex].
Step 1: Evaluate the left-hand side (LHS) of the inequality.
The left-hand side is given by:
[tex]\[ -6(-4 + x) \][/tex]
Substitute [tex]\( x = 10 \)[/tex]:
[tex]\[ -6(-4 + 10) \][/tex]
First, simplify inside the parentheses:
[tex]\[ -4 + 10 = 6 \][/tex]
Now, multiply by [tex]\(-6\)[/tex]:
[tex]\[ -6 \cdot 6 = -36 \][/tex]
So, the left-hand side is [tex]\(-36\)[/tex].
Step 2: Evaluate the right-hand side (RHS) of the inequality.
The right-hand side is given by:
[tex]\[ 120 + 4x \][/tex]
Substitute [tex]\( x = 10 \)[/tex]:
[tex]\[ 120 + 4 \cdot 10 \][/tex]
First, multiply:
[tex]\[ 4 \cdot 10 = 40 \][/tex]
Now, add:
[tex]\[ 120 + 40 = 160 \][/tex]
So, the right-hand side is [tex]\( 160 \)[/tex].
Step 3: Compare the left-hand side and the right-hand side.
We have:
[tex]\[ \text{LHS} = -36 \][/tex]
[tex]\[ \text{RHS} = 160 \][/tex]
Now, compare the two sides:
[tex]\[ -36 < 160 \][/tex]
Since [tex]\(-36\)[/tex] is indeed less than [tex]\(160\)[/tex], the inequality [tex]\( -6(-4 + x) < 120 + 4x \)[/tex] holds true when [tex]\( x = 10 \)[/tex].
Thus, [tex]\( x = 10 \)[/tex] is a solution to the inequality [tex]\( -6(-4 + x) < 120 + 4x \)[/tex].
