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Answer :
- Add $\frac{2}{3}x$ to both sides: $6 < x - 9 + \frac{2}{3}x$.
- Add 9 to both sides: $15 < x + \frac{2}{3}x$.
- Combine $x$ terms: $15 < \frac{5}{3}x$.
- Multiply by $\frac{3}{5}$: $9 < x$, thus $x > 9$.
- The final answer is $\boxed{x>9}$.
### Explanation
1. Understanding the Inequality
We are given the inequality $6 - \\frac{2}{3}x < x - 9$. Our goal is to isolate $x$ on one side of the inequality to find an equivalent expression.
2. Adding $\\frac{2}{3}x$ to Both Sides
First, let's add $\\frac{2}{3}x$ to both sides of the inequality to get rid of the term with $x$ on the left side:$$6 - \\frac{2}{3}x + \\frac{2}{3}x < x - 9 + \\frac{2}{3}x$$This simplifies to:$$6 < x - 9 + \\frac{2}{3}x$$
3. Adding 9 to Both Sides
Next, add 9 to both sides of the inequality to isolate the $x$ terms on the right side:$$6 + 9 < x + \\frac{2}{3}x - 9 + 9$$This simplifies to:$$15 < x + \\frac{2}{3}x$$
4. Combining $x$ Terms
Now, combine the $x$ terms on the right side. We can rewrite $x$ as $\\frac{3}{3}x$, so we have:$$15 < \\frac{3}{3}x + \\frac{2}{3}x$$This simplifies to:$$15 < \\frac{5}{3}x$$
5. Multiplying by $\\frac{3}{5}
To isolate $x$, multiply both sides of the inequality by $\\frac{3}{5}$:$$\\frac{3}{5} \\cdot 15 < \\frac{3}{5} \\cdot \\frac{5}{3}x$$This simplifies to:$$9 < x$$
6. Final Answer
Therefore, the inequality $6 - \\frac{2}{3}x < x - 9$ is equivalent to $x > 9$.
### Examples
Understanding inequalities is crucial in many real-world scenarios, such as budgeting. For example, if you have a certain amount of money and need to ensure your expenses stay below that limit, you're essentially dealing with an inequality. Similarly, in physics, inequalities can help define the range of possible outcomes in experiments, ensuring results stay within acceptable bounds. This problem demonstrates a fundamental skill in manipulating inequalities to solve for an unknown variable, which is applicable in various fields.
- Add 9 to both sides: $15 < x + \frac{2}{3}x$.
- Combine $x$ terms: $15 < \frac{5}{3}x$.
- Multiply by $\frac{3}{5}$: $9 < x$, thus $x > 9$.
- The final answer is $\boxed{x>9}$.
### Explanation
1. Understanding the Inequality
We are given the inequality $6 - \\frac{2}{3}x < x - 9$. Our goal is to isolate $x$ on one side of the inequality to find an equivalent expression.
2. Adding $\\frac{2}{3}x$ to Both Sides
First, let's add $\\frac{2}{3}x$ to both sides of the inequality to get rid of the term with $x$ on the left side:$$6 - \\frac{2}{3}x + \\frac{2}{3}x < x - 9 + \\frac{2}{3}x$$This simplifies to:$$6 < x - 9 + \\frac{2}{3}x$$
3. Adding 9 to Both Sides
Next, add 9 to both sides of the inequality to isolate the $x$ terms on the right side:$$6 + 9 < x + \\frac{2}{3}x - 9 + 9$$This simplifies to:$$15 < x + \\frac{2}{3}x$$
4. Combining $x$ Terms
Now, combine the $x$ terms on the right side. We can rewrite $x$ as $\\frac{3}{3}x$, so we have:$$15 < \\frac{3}{3}x + \\frac{2}{3}x$$This simplifies to:$$15 < \\frac{5}{3}x$$
5. Multiplying by $\\frac{3}{5}
To isolate $x$, multiply both sides of the inequality by $\\frac{3}{5}$:$$\\frac{3}{5} \\cdot 15 < \\frac{3}{5} \\cdot \\frac{5}{3}x$$This simplifies to:$$9 < x$$
6. Final Answer
Therefore, the inequality $6 - \\frac{2}{3}x < x - 9$ is equivalent to $x > 9$.
### Examples
Understanding inequalities is crucial in many real-world scenarios, such as budgeting. For example, if you have a certain amount of money and need to ensure your expenses stay below that limit, you're essentially dealing with an inequality. Similarly, in physics, inequalities can help define the range of possible outcomes in experiments, ensuring results stay within acceptable bounds. This problem demonstrates a fundamental skill in manipulating inequalities to solve for an unknown variable, which is applicable in various fields.
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