To solve the inequality [tex]\( -4 - \frac{1}{5} x > 3 \)[/tex], follow these steps:
1. Isolate the variable term:
Start by moving the constant term [tex]\(-4\)[/tex] to the right side of the inequality. We do this by adding [tex]\(4\)[/tex] to both sides:
[tex]\[
-4 - \frac{1}{5} x + 4 > 3 + 4
\][/tex]
This simplifies to:
[tex]\[
-\frac{1}{5} x > 7
\][/tex]
2. Eliminate the fraction:
To clear the fraction, multiply both sides of the inequality by [tex]\(-5\)[/tex], noting that multiplying by a negative number reverses the inequality sign:
[tex]\[
\left( -\frac{1}{5} x \right) (-5) < 7 (-5)
\][/tex]
Simplifying this gives:
[tex]\[
x < -35
\][/tex]
Therefore, the solution to the inequality [tex]\( -4 - \frac{1}{5} x > 3 \)[/tex] is [tex]\( x < -35 \)[/tex]. This means that [tex]\( x \)[/tex] can take any value less than [tex]\(-35\)[/tex].
In interval notation, the solution is:
[tex]\[
(-\infty, -35)
\][/tex]
Thus, the inequality [tex]\( -4 - \frac{1}{5} x > 3 \)[/tex] holds true for all [tex]\( x \)[/tex] in the interval [tex]\((- \infty, -35)\)[/tex].
