Solve the following inequality for w: 1 - 3(4w - 6) ≤ 9w + 3 - 10w.

Understand the Problem

The question is asking to solve a linear inequality for the variable w and express the answer in its simplest form. The inequality provided includes a combination of linear terms and constants, requiring simplification and manipulation to isolate w.

Answer

$w \geq \frac{16}{11}$

Steps to Solve

Distribute the terms inside the inequality

Start by distributing the $-3$ across the expression $4w - 6$: $$ 1 - 3(4w - 6) \leq 9w + 3 - 10w $$

This simplifies to: $$ 1 - 12w + 18 \leq 9w + 3 - 10w $$

Combine like terms

Next, combine the constant terms on the left side and the variable terms on the right side: $$ (1 + 18) - 12w \leq (9w - 10w + 3) $$

This simplifies to: $$ 19 - 12w \leq -w + 3 $$

Isolate the variable

Now, to isolate $w$, add $12w$ to both sides: $$ 19 \leq 12w - w + 3 $$

This simplifies to: $$ 19 \leq 11w + 3 $$

Subtract constant terms

Next, subtract $3$ from both sides: $$ 19 - 3 \leq 11w $$

This gives us: $$ 16 \leq 11w $$

Divide by the coefficient of w

Finally, divide both sides by $11$ to solve for $w$: $$ \frac{16}{11} \leq w $$

This can also be written as: $$ w \geq \frac{16}{11} $$

The solution to the inequality is

$$ w \geq \frac{16}{11} $$

More Information

The inequality indicates that the value of $w$ must be greater than or equal to $\frac{16}{11}$. This represents a range of values for $w$ starting from approximately 1.45 and extending to positive infinity. Inequalities are useful in various real-world contexts, such as budgeting, scheduling, and resource allocation.

Tips

Failing to correctly distribute negative signs when dealing with subtraction can lead to incorrect terms.
Forgetting to swap the inequality sign when multiplying or dividing by a negative number (not applicable here, but important to remember).
Not combining like terms properly, leading to incorrect equations.

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