To solve the inequality
[tex]\[ -7 \geq \frac{x}{4} - 6, \][/tex]
follow these steps:
1. Isolate the term containing [tex]\( x \)[/tex]:
Start by isolating the [tex]\( \frac{x}{4} \)[/tex] term on one side of the inequality. To do this, add 6 to both sides:
[tex]\[
-7 + 6 \geq \frac{x}{4}.
\][/tex]
2. Simplify:
Simplify the left-hand side:
[tex]\[
-1 \geq \frac{x}{4}.
\][/tex]
3. Eliminate the fraction:
To get rid of the fraction, multiply both sides of the inequality by 4:
[tex]\[
4 \cdot (-1) \geq 4 \cdot \frac{x}{4}.
\][/tex]
4. Simplify:
Simplify both sides:
[tex]\[
-4 \geq x.
\][/tex]
This can also be written as:
[tex]\[
x \leq -4.
\][/tex]
So the solution to the inequality is
[tex]\[ x \leq -4. \][/tex]
In interval notation, we write the solution as:
[tex]\[ (-\infty, -4]. \][/tex]
This means that [tex]\( x \)[/tex] can be any value less than or equal to -4.
