A number, w, is decreased by 6 and the result is multiplied by 4. The final result is greater than the original number. Write and solve an inequality for w.
Answer (1)
Form the inequality: w"> 4 ( w − 6 ) > w .
Distribute: w"> 4 w − 24 > w .
Simplify: 24"> 3 w > 24 .
Solve for w: 8"> w > 8 . The solution is 8}"> w > 8 .
Explanation
Understanding the Problem Let's break down the problem step by step to form an inequality and then solve it.
Translating the First Part We are given that a number, w , is decreased by 6. This can be written as w − 6 .
Multiplying by 4 The result of decreasing w by 6 is then multiplied by 4. This gives us 4 ( w − 6 ) .
Forming the Inequality We are told that the final result is greater than the original number w . So, we can write the inequality as: w"> 4 ( w − 6 ) > w
Distributing Now, let's solve the inequality for w . First, distribute the 4: w"> 4 w − 24 > w
Subtracting w Next, subtract w from both sides of the inequality: w - w"> 4 w − w − 24 > w − w
0"> 3 w − 24 > 0
Adding 24 Now, add 24 to both sides of the inequality: 0 + 24"> 3 w − 24 + 24 > 0 + 24
24"> 3 w > 24
Dividing by 3 Finally, divide both sides by 3: \frac{24}{3}"> 3 3 w > 3 24
8"> w > 8
Final Answer So, the solution to the inequality is 8"> w > 8 . This means that w must be greater than 8.
Examples
Imagine you're managing a small business and you need to determine the minimum number of products you need to sell to make a profit. If each product costs you $6 to produce, and you make $4 profit on each sale after that cost, you want to ensure your total profit is greater than your initial investment. This problem is similar; it helps you find the minimum value (number of products) needed to exceed a certain threshold (initial investment). By solving the inequality, you determine the minimum number of sales required to achieve your goal.
