Solve the following inequality for v: 8 - (9v - 2) ≥ v - 6 - 5.
Understand the Problem
The question is asking to solve a linear inequality involving the variable 'v'. The task is to simplify the inequality and express the solution clearly.
Answer
\( v \leq \frac{21}{10} \)
Answer for screen readers
The solution to the inequality is ( v \leq \frac{21}{10} ).
Steps to Solve
- Distribute and simplify the left side of the inequality
Start with the original inequality:
$$ 8 - (9v - 2) \geq v - 6 - 5 $$
Distributing the negative sign:
$$ 8 - 9v + 2 \geq v - 6 - 5 $$
Then combine like terms:
$$ 10 - 9v \geq v - 11 $$
- Isolate the variable term
Next, we want to get all terms involving $v$ on one side. Add $9v$ to both sides:
$$ 10 \geq v + 9v - 11 $$
$$ 10 \geq 10v - 11 $$
Now add $11$ to both sides:
$$ 10 + 11 \geq 10v $$
$$ 21 \geq 10v $$
- Solve for v
Now, divide both sides by $10$:
$$ \frac{21}{10} \geq v $$
This can also be written as:
$$ v \leq \frac{21}{10} $$
The solution to the inequality is ( v \leq \frac{21}{10} ).
More Information
The solution indicates that the values of ( v ) can be any number less than or equal to ( \frac{21}{10} ) (which is equal to 2.1). This means that ( v ) can take any value up to 2.1.
Tips
- Forgetting to distribute the negative sign correctly.
- Incorrectly moving terms across the inequality sign without changing the direction, usually when multiplying or dividing by a negative number.
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