To solve the inequality [tex]\(x - 4 > 4(x - 1)\)[/tex], follow these steps:
1. Expand and Simplify:
First, distribute the 4 on the right-hand side:
[tex]\[
x - 4 > 4(x - 1) \implies x - 4 > 4x - 4 \cdot 1
\implies x - 4 > 4x - 4
\][/tex]
2. Combine Like Terms:
Next, we need to collect all [tex]\(x\)[/tex]-terms on one side and constants on the other side. Start by subtracting [tex]\(x\)[/tex] from both sides:
[tex]\[
x - 4 - x > 4x - x - 4 \implies -4 > 3x - 4
\][/tex]
Then, add 4 to both sides:
[tex]\[
-4 + 4 > 3x - 4 + 4 \implies 0 > 3x
\][/tex]
3. Isolate [tex]\(x\)[/tex]:
Finally, divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[
0 > 3x \implies 0 > x
\][/tex]
Rewriting this, we get:
[tex]\[
x < 0
\][/tex]
The solution to the inequality [tex]\(x - 4 > 4(x - 1)\)[/tex] is:
[tex]\[
x < 0
\][/tex]
In interval notation, the solution is:
[tex]\[
(-\infty, 0)
\][/tex]
