Solve: Four Times A Number Less Than 56

Ever wondered how to tackle word problems in math? Let's dive into one together! Today, we're going to explore a problem that asks: Four times a number is less than 56. What are the possible values of that number? This seemingly simple question opens up a world of possibilities and helps us understand inequalities. When we're faced with a problem like this, the first step is always to translate the words into mathematical language. "Four times a number" can be represented as 4x, where x is our unknown number. The phrase "is less than" translates to the inequality symbol <. And finally, "56" is simply the number 56. So, putting it all together, we get the inequality: 4x < 56. Now, the goal is to isolate x to find out what values it can take. To do this, we need to get rid of the 4 that's multiplying x. The opposite of multiplication is division, so we'll divide both sides of the inequality by 4. Remember, when you perform an operation on one side of an inequality, you must do the same on the other side to keep it balanced. So, 4x / 4 becomes x, and 56 / 4 equals 14. Therefore, our inequality simplifies to x < 14. This means that the number x can be any value that is less than 14. It could be 13, 10, 0, -5, or even a decimal like 13.999! The possibilities are infinite as long as the number is strictly smaller than 14. This is a fundamental concept in algebra, and understanding how to translate word problems into equations or inequalities is a key skill. Keep practicing, and you'll become a pro at solving these in no time!

Understanding Inequalities and Algebraic Expressions

Let's dig a little deeper into the world of inequalities and algebraic expressions, which are central to solving problems like "Four times a number is less than 56. What are the possible values of that number?" An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x, y, or z), and operators (like addition, subtraction, multiplication, and division). In our problem, 4x is an algebraic expression representing "four times a number." The variable x stands for the unknown number we're trying to find. An inequality, on the other hand, is a statement that compares two expressions using symbols like <, >, ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike an equation (which uses =), an inequality doesn't state that two things are exactly equal; instead, it shows a relationship of being greater than or less than. Our problem gives us the inequality 4x < 56. Solving an inequality involves isolating the variable, just like solving an equation. However, there's a crucial rule to remember with inequalities: if you multiply or divide both sides by a negative number, you must reverse the inequality sign. For example, if we had -2x < 10, dividing by -2 would give us x > -5 (the < sign flips to >). In our specific case, 4x < 56, we divide by a positive number (4), so the inequality sign remains the same. Dividing both sides by 4, we get x < 14. This result tells us that any number smaller than 14 will satisfy the original condition. Think about it: if you pick a number less than 14, say 10, and multiply it by 4, you get 40, which is indeed less than 56. If you pick a number greater than or equal to 14, say 15, and multiply it by 4, you get 60, which is not less than 56. So, the solution x < 14 accurately captures all the possible values for our number. This process highlights the power of algebra in representing and solving real-world scenarios, from simple word problems to complex scientific formulas. It's all about translating the language of the problem into the precise language of mathematics.

Exploring the Options: Why 'x < 14' is Key

When we solve the inequality derived from the statement "Four times a number is less than 56," we arrive at x < 14. This solution is not just a single number; it represents an infinite set of possibilities. The question asks for the possible values of that number, and x < 14 encapsulates all of them. Let's break down why this is the correct representation and how it relates to the given multiple-choice options, although the options provided in the prompt (A, B, C, D) don't directly include the precise answer x < 14. They seem to be referencing a threshold value, which in this case is 14. Let's assume the options were meant to test understanding around the number 14. We found that x must be strictly less than 14. This means numbers like 13, 10, 0, -100, or any decimal below 14 are valid. The number 14 itself is not included, because if x = 14, then 4x = 4 * 14 = 56, which is not less than 56. It is equal to 56. Therefore, any value of x that is greater than 14 is also incorrect. For instance, if x = 15, 4x = 4 * 15 = 60, which is greater than 56. So, the condition x < 14 is the only one that satisfies the original problem statement. If we were to look at the provided options: A. $x>12$, B. $x<12$, C. $x<16$, D. $x>16$. None of these perfectly match our derived solution x < 14. However, if we had to choose the closest or a potentially intended answer based on a misunderstanding or simplification, option C, x < 16, is closer in the sense of indicating a range below a certain number. But mathematically, x < 14 is the precise and only correct answer. It's important to be precise in mathematics. The inequality x < 14 tells us that the number can be any real number that falls to the left of 14 on the number line. The difference between x < 14 and x < 16 is significant; x < 16 would include numbers like 14 and 15, which we've shown do not satisfy the condition. Similarly, options A and D suggest values greater than a certain number, which is the opposite of what our problem requires. This reinforces the importance of carefully translating word problems and performing the correct algebraic manipulations to arrive at the exact solution.

Conclusion: Mastering the Math

In conclusion, the problem "Four times a number is less than 56. What are the possible values of that number?" leads us directly to the inequality 4x < 56. By applying basic algebraic principles, specifically dividing both sides by 4, we confidently determine that the possible values of the number x must be less than 14. This is represented mathematically as x < 14. It's a powerful demonstration of how simple English phrases can be converted into precise mathematical statements, allowing us to solve for unknown quantities. Remember, the key steps are translating the words into an inequality and then isolating the variable. Always double-check your work, especially when dealing with inequalities, to ensure you've applied the rules correctly. Whether you're just starting with algebra or looking to refine your skills, practicing these types of problems will build a strong foundation for more advanced mathematical concepts. Keep exploring, keep questioning, and keep learning!

For further exploration into the fascinating world of mathematics and inequalities, you might find the resources at Khan Academy to be incredibly helpful. They offer a vast array of free lessons and practice exercises covering topics from basic arithmetic to advanced calculus. You can visit them at www.khanacademy.org.