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Chapter 2: Problem 13

For exercises 9-36, (a) solve. (b) check the direction of the inequality sign. $$ -p+12>3 $$

Short Answer

Expert verified
p < 9

Step by step solution

01

- Isolate the variable term

Subtract 12 from both sides of the inequality to isolate the variable term:\[ -p + 12 - 12 > 3 - 12 \]\[ -p > -9 \]
02

- Solve for the variable

Divide both sides of the inequality by -1 to solve for p, and remember to reverse the inequality sign when dividing by a negative:\[ -p \, / \, -1 < -9 \, / \, -1 \]\[ p < 9 \]
03

- Check the direction of the inequality sign

When dividing by a negative number, the inequality sign must be reversed. Verify this by substituting a value less than 9 (e.g., 8) into the original inequality to see if it holds true:\[ -p + 12 > 3 \Rightarrow -8 + 12 > 3 \Rightarrow 4 > 3 \]

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Key Concepts

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

Isolate the Variable
To solve any inequality, your first goal should be to isolate the variable on one side of the inequality.

Here's how to do it: Start with the inequality given: \[ -p + 12 > 3 \] You need to get rid of the +12 on the left side. To do this, subtract 12 from both sides: \[ -p + 12 - 12 > 3 - 12 \] This simplifies to: \[ -p > -9 \] Now, the variable term is isolated on one side of the inequality.
Reverse Inequality Sign
When solving inequalities, it's essential to remember that if you multiply or divide by a negative number, you must reverse the inequality sign.

Let's look at our isolated inequality: \[ -p > -9 \] To solve for \( p \), divide both sides by -1: \[ \frac{-p}{-1} < \frac{-9}{-1} \] Notice how the inequality sign reverses from > to <: \[ p < 9 \] This step is crucial because failing to reverse the inequality sign when dividing by a negative can lead to incorrect solutions.
Substitute and Verify
After solving the inequality, it is always a good idea to substitute a value back into the original inequality to verify your solution.

In our case, we have: \[ p < 9 \] Let's choose a value less than 9, say 8, and substitute it back into the original inequality: \[ -p + 12 > 3 \] Replace \( p \) with 8: \[ -8 + 12 > 3 \] Simplify the left side: \[ 4 > 3 \] This is true, which confirms that our solution \( p < 9 \) is correct. Substituting values helps us verify that the inequality holds true and gives us confidence in our solution.

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Most popular questions from this chapter

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