Three Times A Number W Is Less Than 18

Three Times a Number w is Less Than 18: A Complete Guide to Solving and Understanding

The phrase “three times a number w is less than 18” is a classic example of a verbal mathematical statement that translates directly into an algebraic inequality. Understanding how to convert such words into symbols and solve for the unknown variable is a foundational skill in algebra. This concept isn’t just about passing a test; it’s a practical tool for modeling real-world situations where quantities have limits or constraints. This guide will walk you through every step, from interpretation to solution, and show you why this simple inequality opens the door to more complex problem-solving.

Understanding the Statement: From Words to Symbols

The first and most critical step is accurately translating the English sentence into a mathematical expression. Let’s dissect it piece by piece:

“a number w”: This introduces our variable, represented by the letter w. It stands for an unknown quantity we need to find or describe.
“three times a number w”: The word “times” means multiplication. So, this part translates to 3 * w or simply 3w.
“is less than”: This is a key inequality phrase. It does not mean “equals.” The symbol for “less than” is <.
“18”: This is our constant, the number we are comparing against.

Putting it all together, “three times a number w is less than 18” becomes the algebraic inequality: 3w < 18

This inequality is not a statement of equality but a relationship. It tells us that the value of 3w must be smaller than 18. Our goal is to find all possible values of w that make this statement true.

Solving the Inequality: A Step-by-Step Process

Solving for w means isolating the variable on one side of the inequality sign. The process is remarkably similar to solving a standard equation, with one crucial rule to remember.

Step 1: Write the Inequality We start with our translated statement: 3w < 18

Step 2: Isolate the Variable The variable w is currently multiplied by 3. To undo this operation, we must perform the inverse operation: division. We will divide both sides of the inequality by 3. (3w) / 3 < 18 / 3

Step 3: Simplify On the left, the 3s cancel out, leaving just w. On the right, 18 divided by 3 is 6. w < 6

The Crucial Rule About Inequalities: When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol (e.g., < becomes >). In this case, we divided by a positive 3, so the < symbol remains unchanged. This is why w < 6 is our final answer.

Interpreting the Solution: What Does w < 6 Mean?

The solution w < 6 is not a single answer but an entire solution set. It describes all possible values for w.

In Set Notation: { w | w < 6 } which reads as “the set of all w such that w is less than 6.”
In Interval Notation: (-∞, 6) This means all real numbers from negative infinity up to (but not including) 6.
On a Number Line: We represent this with an open circle at 6 (because 6 itself is not a solution—3*6 = 18, which is not less than 18) and a shaded line extending endlessly to the left, towards smaller numbers.

Key Insight: Any number smaller than 6 is a solution. Examples include 5, 0, -2, -100, and 5.999. The number 6 is the boundary but is not included in the solution.

Real-World Applications: Where This Pops Up

This type of inequality models countless practical scenarios:

Budgeting: “You have $18. Each item costs $3. How many items (w) can you buy?” You cannot spend exactly $18 or more. So, 3w < 18 means you can buy at most 5 items (since w must be a whole number in this context, the integer solutions are 0, 1, 2, 3, 4, 5).
Weight Limits: “A elevator can carry a maximum of 18 tons. Each crate weighs 3 tons. How many crates (w) can be safely loaded?” Again, 3w < 18 applies.
Time Management: “A task takes 3 hours per unit. You have less than 18 hours available. How many units (w) can you complete?”
Science: In chemistry, if a reaction requires 3 moles of substance A for every mole of product, and you have less than 18 moles of A, the inequality 3w < 18 (where w is moles of product) limits your yield.

Common Mistakes and How to Avoid Them

Confusing “Less Than” with “Less Than or Equal To”: The phrase “is less than” strictly means <. If the problem said “is less than or equal to 18,” the inequality would be 3w ≤ 18, and the solution would be w ≤ 6, including 6. Always read the wording precisely.
Forgetting to Reverse the Inequality Sign: This only happens when multiplying or dividing by a negative. A common trick is to test a number. If you incorrectly solved -2w < 6 as w < -3, plug in w = -4 (which is less than -3). -2*(-4) = 8, and 8 is not less than 6. The correct solution w > -3 works: -2*(-2) = 4 < 6.
Interpreting the Solution as a Single Integer: Unless the context specifies w must be a whole number (like counting items), w < 6 includes all decimals and fractions less than 6. In the crate example, you can’t load a fraction of a crate, so you’d interpret the solution as the integers w = 0, 1, 2, 3, 4, 5.

FAQ: Addressing Your Follow-Up Questions

Q1: What if the problem was “three times a number w is no more than 18”? “No more than” means “less than or equal to.” The inequality becomes 3w ≤ 18, and

Q1: What if the problem was “three times a number w is no more than 18”?
"No more than" means "less than or equal to." The inequality becomes 3w ≤ 18, and the solution is w ≤ 6. On a number line, this is represented with a closed circle at 6 (indicating inclusion) and shading extending to the left. The boundary point is included because 3*6 = 18 satisfies "no more than 18."

Q2: How do I graph solutions involving "greater than" or "greater than or equal to"?
The direction of shading flips. For example:

w > 6: Open circle at 6, shading to the right (towards larger numbers).
w ≥ 6: Closed circle at 6, shading to the right.
w ≤ 6: Closed circle at 6, shading to the left.
w < 6: Open circle at 6, shading to the left.
Remember: The shading always points away from the excluded region (the "less than" side for < or ≤, the "greater than" side for > or ≥).

Conclusion

Inequalities like 3w < 18 are fundamental tools for expressing constraints and boundaries in mathematics and the real world. They define ranges of solutions rather than single points, capturing the essence of "less than," "greater than," and their inclusive counterparts. Understanding how to translate words into symbols (<, >, ≤, ≥), solve the inequality by isolating the variable, and represent the solution graphically on a number line is crucial. Key pitfalls to avoid include misinterpreting boundary conditions (strict inequality vs. inclusive inequality) and forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Whether budgeting, managing weight limits, or planning time, inequalities provide the precise language needed to describe limitations and possibilities, making them indispensable for logical problem-solving across countless disciplines. Mastering them unlocks the ability to navigate the boundaries inherent in quantitative situations.