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Chapter 6: Problem 35

Solve the inequality and graph its solution. \(p-12 \geq-1\)

Short Answer

Expert verified
The solution to the inequality is \(p \geq 11\). It is graphed with a filled-in circle at 11 and an arrow pointing to the right on a number line.

Step by step solution

01

Set up the inequality

Our initial inequality is \(p - 12 \geq -1\). The goal now is to solve the inequality for \(p\).
02

Solve the inequality

To solve for \(p\), add 12 to both sides of the equation. So, \(p = -1 + 12\), which simplifies to \(p \geq 11\).
03

Graph the solution

The solution to this inequality is \(p \geq 11\). This means any number larger or equal to 11 is a solution to the inequality. On a number line, this would be graphed with a filled-in circle at 11 (since \(p\) can equal 11) and an arrow pointing to the right (since all numbers larger than 11 are solutions).

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Key Concepts

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

graphing solutions
When we solve inequalities, we often need to visualize the solutions to better understand them. Graphing is a great way to do this. In the example given, we solved the inequality \(p - 12 \geq -1\) and found that \(p \geq 11\). To graph this:
  • Start by drawing a number line, a simple horizontal line with evenly spaced numbers.
  • Locate the number 11 on this line, as this is the boundary which the solutions cannot go below.
  • Place a filled-in circle (or dot) on 11 to indicate that 11 is part of the solution set, since \(p\) can equal 11.
  • Draw an arrow from the circle extending to the right, indicating that the solution includes all numbers greater than 11.
This graphical representation helps to quickly see which values \(p\) can take, making it an efficient tool for interpreting the results of inequalities.

Keep in mind:
  • A filled-in circle is used because the inequality sign \(\geq\) includes the number itself.
  • If the inequality were \(p > 11\), you would use an open circle to show that 11 itself is not included.
one-variable inequalities
One-variable inequalities involve expressions where only one variable is present. In solving these, the main aim is to isolate the variable on one side to find its possible values. For the inequality \(p - 12 \geq -1\), our goal was to find out what \(p\) could be.Steps to solve:
  • Analyze the given inequality and decide on what operations are needed to isolate the variable.
  • Perform these operations on both sides. For \(p - 12 \geq -1\), we added 12 to both sides.
  • The result, \(p \geq 11\), tells us that \(p\) must be 11 or greater.
Understanding these steps allows you to handle more complex inequalities later on. Remember, solving inequalities is about maintaining balance; whatever you do to one side, you must do to the other.

These skills are important in math and beyond, as they form the basis of reasoning through problems with conditions or limits.
algebraic expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. In the context of inequalities, such expressions help describe relationships between different quantities.For the given inequality \(p - 12 \geq -1\), the expression \(p - 12\) represents the value of \(p\) reduced by 12. Here's how to work with such expressions:
  • Identify the variable and constants in the expression. Here, \(p\) is the variable and 12 is a constant.
  • To solve it, you need to perform algebraic operations that help isolate the variable.
  • Add or subtract constants on both sides or multiply/divide by a number (be careful with multiplication or division involving negative numbers, as they reverse the inequality sign).
Understanding algebraic expressions is crucial since they form the foundation for modeling real-world scenarios mathematically.

These skills help in breaking down complex problems into manageable parts, making them an essential tool in a student鈥檚 math toolkit.

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Most popular questions from this chapter

Evaluate the expression. $$\frac{37+65+89+72+82}{5}$$

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