Certainly! Let's solve the inequality step-by-step:
Given the inequality:
[tex]\[
\frac{1}{2}(x-2) > \frac{1}{4}(7x-2)
\][/tex]
### Step 1: Clear the Fractions
To eliminate the fractions, we can multiply both sides of the inequality by 4 (the least common multiple of the denominators 2 and 4):
[tex]\[
4 \cdot \frac{1}{2}(x-2) > 4 \cdot \frac{1}{4}(7x-2)
\][/tex]
This simplifies to:
[tex]\[
2(x-2) > 7x-2
\][/tex]
### Step 2: Distribute the Constants
Next, we distribute the constants inside the parentheses:
[tex]\[
2x - 4 > 7x - 2
\][/tex]
### Step 3: Arrange All Terms Involving [tex]\( x \)[/tex] on One Side
To isolate the [tex]\( x \)[/tex] terms on one side of the inequality, we can subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[
-4 > 5x - 2
\][/tex]
Next, add 2 to both sides to further isolate the [tex]\( x \)[/tex] term:
[tex]\[
-2 > 5x
\][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
To solve for [tex]\( x \)[/tex], divide both sides by 5:
[tex]\[
\frac{-2}{5} > x
\][/tex]
This can be rewritten as:
[tex]\[
x < \frac{-2}{5}
\][/tex]
### Final Answer
Therefore, the solution to the inequality is:
[tex]\[
x < -0.4
\][/tex]
This inequality indicates that [tex]\( x \)[/tex] must be less than [tex]\(-0.4\)[/tex] to satisfy the given condition.
