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Chapter 1: Problem 16

Graph all solutions on a number line and provide the corresponding interval notation. ?2a+5>5

Short Answer

Expert verified
Graph: Open circle at 0, arrow to the right. Interval: \((0, \infty)\).

Step by step solution

01

Simplify the Inequality

Start by simplifying the inequality. Given: \[ 2a + 5 > 5 \] Subtract 5 from both sides to isolate the term with the variable:\[ 2a + 5 - 5 > 5 - 5 \] Which simplifies to:\[ 2a > 0 \]
02

Solve for the Variable

Next, solve for \( a \) by dividing both sides of the inequality by 2:\[ \frac{2a}{2} > \frac{0}{2} \] This simplifies to:\[ a > 0 \] This means any number greater than zero is a solution.
03

Graph the Solution on a Number Line

To graph the solution \( a > 0 \), draw a number line. Place an open circle on zero to indicate that zero is not included. Then draw a line or arrow extending to the right, reflecting that all numbers greater than zero are solutions.
04

Express in Interval Notation

Finally, express the solution \( a > 0 \) in interval notation. Since it includes all numbers greater than zero, the interval is:\( (0, \infty) \) The parenthesis indicates that zero is not included, and the infinity symbol indicates it goes infinitely large.

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Key Concepts

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

Graphing on a Number Line
Graphing inequalities on a number line helps us visually represent the range of solutions for an inequality. It鈥檚 like creating a map for numbers that satisfy the given condition. Here's how to do it:
  • Identify the inequality's condition, like if it鈥檚 greater than (>), less than (<), or includes equal to (鈮� or 鈮�).
  • Find the critical point(s) from the solved inequality. In the exercise, we found that the critical point is 0.
  • Use a number line and mark the critical point. For inequalities without 'equal to', use an open circle. If the inequality includes 'equal to', use a closed circle.
  • Draw an arrow or line extending from the circle in the direction specified by the inequality symbol. For example, for '> 0', draw an arrow extending to the right from zero.
This method clearly illustrates which numbers are solutions, making it easier to understand and confirm your answer.
Solving Inequalities
Solving inequalities is similar to solving equations, with some key differences. Here鈥檚 how to approach them:
  • First, simplify the inequality by performing operations like addition, subtraction, multiplication, or division, as you would with an equation. For instance, we started with \( 2a + 5 > 5 \).
  • Isolate the variable on one side of the inequality. Often, this involves getting rid of constants on the same side as the variable, as seen with \( 2a > 0 \).
  • When dividing or multiplying both sides by a negative number, remember to flip the inequality sign. For example, if dividing by -1, '>' becomes '<'.
In the given exercise, we divided both sides by 2 to isolate \( a \), leading us to \( a > 0 \). The goal is to find values of \( a \) that satisfy the simplified inequality, marking all those numbers as solutions.
Interval Notation
Interval notation offers a concise way to express a range of solutions from an inequality. It is a handy tool to indicate which numbers are solutions without having to draw them. Here鈥檚 how it works:
  • You use parentheses \( ( ) \) to show that endpoints are not included in the solution set, much like an open circle on a number line.
  • Use brackets \( [ ] \) if the endpoints are included, akin to a closed circle.
  • The two numbers in the notation signify the smallest and largest numbers in the solution set, respectively. For example, in the inequality \( a > 0 \), zero is not included, so we write \( (0, \infty) \).
  • The infinity symbol (often written as \( \infty \)) indicates the solution continues indefinitely in a positive or negative direction.
This form provides a clear snapshot of the solution set, making it easier to communicate and understand.

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