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Chapter 2: Problem 57

For exercises 53-62, (a) clear the fractions or decimals and solve. (b) check the direction of the inequality sign. $$ 0.2 w+5<8.6 $$

Short Answer

Expert verified
The solution is \( w < 18 \).

Step by step solution

01

- Clear Decimals

To clear decimals in the inequality, multiply both sides by 10 (the smallest power of 10 that eliminates the decimal). So, we get: \[ 10(0.2w + 5) < 10(8.6) \] This simplifies to: \[ 2w + 50 < 86 \]
02

- Isolate the Variable

Subtract 50 from both sides to begin isolating the variable: \[ 2w + 50 - 50 < 86 - 50 \] This simplifies to: \[ 2w < 36 \]
03

- Solve for the Variable

Divide both sides by 2 to solve for the variable: \[ \frac{2w}{2} < \frac{36}{2} \] This simplifies to: \[ w < 18 \]
04

- Verify the Inequality Direction

Check the direction of the inequality sign to ensure it is correct. Since we multiplied and divided by positive numbers, the direction of the inequality remains the same. Therefore, the solution is: \[ w < 18 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Clearing Decimals
First, we need to eliminate decimals to simplify the solving process. This involves transforming the decimal equations into whole numbers. In our example, the equation is: \0.2w + 5 < 8.6\.

We multiply every term by 10, the smallest power of 10 that eliminates the decimal points. By doing this, we get:
\[10(0.2w + 5) < 10(8.6)\]

This simplifies to:
\[2w + 50 < 86\]

By clearing the decimals, we transformed the inequality into a simpler form with whole numbers, making it easier to work on and solve.
Isolating the Variable
Next, we need to isolate the variable, which means getting the variable term alone on one side of the inequality. Let's start with our simplified equation:
\[2w + 50 < 86\]

To isolate the variable term, we need to eliminate the constant term on the same side. We do this by subtracting 50 from both sides:
\[2w + 50 - 50 < 86 - 50\]

This simplifies to:
\[2w < 36\]

Now, we divide both sides by 2 to get \( w \) alone:
\[\frac{2w}{2} < \frac{36}{2}\]

This finally simplifies to:
\[w < 18\]

Thus, we have successfully isolated the variable, providing us with the solution.
Inequality Direction
The direction of the inequality sign is vital in maintaining the correct relationship between the expressions. The general rule is:
  • If you multiply or divide both sides of an inequality by a positive number, the direction of the inequality remains the same.
  • If you multiply or divide both sides by a negative number, the direction flips.
Let's review our steps to ensure the inequality direction was maintained correctly:

When we cleared the decimals by multiplying by 10, we used a positive number, and the inequality direction did not change.
When we subtracted 50, it was just standard subtraction without direction impact.
When we divided by 2, it was a positive number, so again no change in direction.

Thus, the inequality remains \( w < 18 \). Properly handling the direction of the inequality sign ensures the accuracy of our solution.

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