Solve the following inequality:
minus, 7, minus, 9, x, is less than or equal to, minus, one half, x, minus, left parenthesis, minus, 8, x, plus, 1, right parenthesis
−7−9x≤
− 
2
1
​
 x−(−8x+1)

The inequality you provided is:

\[
-7 - 9x \leq -\frac{1}{2}x - \left( -8x + 1 \right)
\]

### Step 1: Simplify the expression on the right-hand side
Expand the parentheses on the right side:

\[
- \left( -8x + 1 \right) = 8x - 1
\]

Thus, the inequality becomes:

\[
-7 - 9x \leq -\frac{1}{2}x + 8x - 1
\]

### Step 2: Combine like terms on the right-hand side
Add \(-\frac{1}{2}x\) and \(8x\):

\[
-\frac{1}{2}x + 8x = \frac{15}{2}x
\]

Now, the inequality becomes:

\[
-7 - 9x \leq \frac{15}{2}x - 1
\]

### Step 3: Move all terms involving \(x\) to one side and constants to the other side
Add \(9x\) to both sides:

\[
-7 \leq \frac{15}{2}x + 9x - 1
\]

Convert \(9x\) to a fraction with a denominator of 2:

\[
9x = \frac{18}{2}x
\]

Now, combine the \(x\)-terms on the right-hand side:

\[
\frac{15}{2}x + \frac{18}{2}x = \frac{33}{2}x
\]

So the inequality becomes:

\[
-7 \leq \frac{33}{2}x - 1
\]

### Step 4: Move the constant term to the left-hand side
Add 1 to both sides:

\[
-7 + 1 \leq \frac{33}{2}x
\]

This simplifies to:

\[
-6 \leq \frac{33}{2}x
\]

### Step 5: Solve for \(x\)
Multiply both sides by \(\frac{2}{33}\) to isolate \(x\):

\[
x \geq \frac{-6 \cdot 2}{33} = \frac{-12}{33} = \frac{-4}{11}
\]

### Final Solution:
\[
x \geq \frac{-4}{11}
\]

The solution to the inequality is:

\[
x \geq \frac{-4}{11}
\]

---

Would you like more details on any part of this solution? Here are some related questions to explore further:

1. How do you solve inequalities with fractions?
2. What are the steps for solving multi-step inequalities?
3. How do you handle negative signs in inequalities?
4. What happens if you multiply or divide both sides of an inequality by a negative number?
5. How can you graph the solution to this inequality on a number line?

**Tip**: Always remember that when you multiply or divide an inequality by a negative number, you must flip the inequality sign!

Solve the following inequality: −7−9x ≤ −(1/2)x − (−8x + 1)

Discover how to solve the linear inequality −7 − 9x ≤ −(1/2)x − (−8x + 1) using step-by-step algebraic techniques. Learn key concepts such as combining like terms, applying the distributive property, and handling inequalities involving fractions. Suitable for students in Grades 9-10.

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