Solve 5x< 12. Write the improper fraction in your answer as a mixed number. Make sure to write
The Correct answer and Explanation is:
To solve the inequality 5x<125x < 125x<12, follow these steps:
Step 1: Isolate the variable
Since the inequality is 5x<125x < 125x<12, the goal is to isolate xxx. To do that, divide both sides of the inequality by 5:x<125x < \frac{12}{5}x<512
Step 2: Convert the improper fraction into a mixed number
The fraction 125\frac{12}{5}512 is an improper fraction because the numerator (12) is greater than the denominator (5). To convert it into a mixed number:
- Divide 12 by 5. The quotient is 2, and the remainder is 2.
- So, 125=225\frac{12}{5} = 2 \frac{2}{5}512=252.
Thus, the inequality becomes:x<225x < 2 \frac{2}{5}x<252
Final Answer:
The solution to the inequality is x<225x < 2 \frac{2}{5}x<252.
Explanation:
- Understanding the inequality: Inequalities are similar to equations, but instead of having an equal sign, they use symbols such as <<<, >>>, ≤\leq≤, or ≥\geq≥. The inequality 5x<125x < 125x<12 means that we are looking for all values of xxx that, when multiplied by 5, result in a product less than 12.
- Dividing to isolate xxx: Since xxx is multiplied by 5, we perform the inverse operation, which is division, to isolate xxx. Dividing both sides of the inequality by 5 yields x<125x < \frac{12}{5}x<512. It’s important to note that when dividing or multiplying both sides of an inequality by a positive number, the direction of the inequality remains the same.
- Converting an improper fraction: An improper fraction has a numerator greater than its denominator, like 125\frac{12}{5}512. To convert it into a mixed number, divide the numerator by the denominator to get the quotient (which is the whole number part) and the remainder (which becomes the numerator of the fractional part).
- Solution interpretation: The solution x<225x < 2 \frac{2}{5}x<252 means that xxx can take any value less than 2252 \frac{2}{5}252, but not equal to it. For example, values like 2, 1, 0, or negative numbers all satisfy this inequality. In terms of graphing, it would represent all values to the left of 2252 \frac{2}{5}252 on a number line.