Sure, let's solve the inequality step by step.
### Given Inequality:
[tex]\[ -\frac{3}{4} x - 6 \leq 18 \][/tex]
### Step 1: Isolate the term involving [tex]\( x \)[/tex]
First, we need to get rid of the constant term on the left-hand side. We can do this by adding 6 to both sides of the inequality:
[tex]\[ -\frac{3}{4} x - 6 + 6 \leq 18 + 6 \][/tex]
[tex]\[ -\frac{3}{4} x \leq 24 \][/tex]
### Step 2: Eliminate the fraction
Now, we need to eliminate the fraction. We can do this by multiplying both sides by the reciprocal of [tex]\(-\frac{3}{4}\)[/tex], which is [tex]\(-\frac{4}{3}\)[/tex]. When we multiply both sides by a negative number, we must reverse the inequality sign:
[tex]\[ x \geq 24 \times \left(-\frac{4}{3}\right) \][/tex]
[tex]\[ x \geq 24 \times -\frac{4}{3} \][/tex]
[tex]\[ x \geq -32 \][/tex]
So, the inequality simplifies to:
[tex]\[ x \geq -32 \][/tex]
### Interpretation with the given conditions:
Now we analyze the given conditions to determine which ones satisfy [tex]\( x \geq -32 \)[/tex].
- [tex]\( x \geq -32 \)[/tex]
- [tex]\( x \geq -18 \)[/tex]
- [tex]\( x \geq -16 \)[/tex]
- [tex]\( x \geq -9 \)[/tex]
Since [tex]\( x \geq -32 \)[/tex] encompasses all the other conditions (as they are greater than [tex]\(-32\)[/tex]), the broadest solution set for [tex]\( x \)[/tex] is:
[tex]\[ x \geq -32 \][/tex]
Hence, the solution to the inequality and the appropriate range from the given conditions is:
[tex]\[ x \geq -32 \][/tex]
So, the inequality [tex]\( -\frac{3}{4} x - 6 \leq 18 \)[/tex] holds true for [tex]\( x \geq -32 \)[/tex].
