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Chapter 2: Problem 23

Solve each inequality. Write the solution set in interval notation and graph it. $$ x+2>5 $$

Short Answer

Expert verified
The solution set in interval notation is \((3, \infty)\) and its graph shows an open circle at 3 and shading to the right.

Step by step solution

01

Understand the Inequality

In this inequality, we need to find the values of \( x \) that make \( x + 2 > 5 \) true. Essentially, we are solving for \( x \) to determine what numbers satisfy the condition \( x + 2 \) being greater than 5.
02

Isolate the Variable

To isolate \( x \), we need to subtract 2 from both sides of the inequality. By performing the subtraction, we get:\[(x + 2) - 2 > 5 - 2\]This simplifies to:\[x > 3\]
03

Write the Solution Set in Interval Notation

The solution \( x > 3 \) means all numbers greater than 3. In interval notation, this is expressed as \((3, \infty)\), indicating that the solution includes all numbers greater than 3, up to infinity, but not including 3 itself.
04

Graph the Solution

To graph \( x > 3 \), draw a number line and place an open circle at 3 to indicate that 3 is not included in the solution set. Then, shade the region to the right of 3 to represent all numbers greater than 3.

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Key Concepts

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

Solving Inequalities
Inequalities are like equations, but instead of showing equal values, they establish a relationship of greater than or less than. When solving an inequality like \( x + 2 > 5 \), your goal is to find all possible values for \( x \) that make the statement true. This process involves isolating \( x \) on one side, similar to solving equations.

For the given inequality:
  • Start by isolating \( x \). This is done by performing inverse operations, such as adding or subtracting the same value from both sides.
  • Subtract 2 from both sides of the inequality \( x + 2 > 5 \) to get \( x > 3 \).
  • The inequality \( x > 3 \) tells us that any number greater than 3 will satisfy the inequality.
This step-by-step approach ensures that we stay within the bounds of algebraic manipulation rules. Remember, unlike equations, flipping the inequality sign happens only when multiplying or dividing by a negative number.
Interval Notation
Interval notation is a concise way to express all the numbers that satisfy an inequality. For the solution \( x > 3 \), we use interval notation to show the range of possible values.

  • In interval notation, parentheses \( () \) represent values that are not included in the solution set.
  • Brackets \( [] \) would be used for values that are included, but in this inequality, 3 itself is not part of the solution. So, we use a parenthesis.
  • For \( x > 3 \), the interval notation is \((3, \infty)\).
  • The symbol \( \infty \) indicates that the set extends indefinitely in the positive direction, beyond any finite number. It's important to pair it with a parenthesis because infinity is not a number we can reach or include.
This notation is particularly useful because it allows us to communicate complex sets in a straightforward manner.
Graphing Inequalities
Graphing inequalities on a number line provides a visual representation of the solution set. This helps you see which numbers are included and excluded from the solution. Let's graph \( x > 3 \).

  • Draw a horizontal line, resembling a number line.
  • Locate the number 3 on the line and draw an open circle around it. The open circle signifies that 3 itself is not included in the solution.
  • Shade the area to the right of the circle to show all numbers greater than 3.
This shading technique efficiently shows the set of all numbers that \( x \) can take in the inequality \( x > 3 \). It communicates the same information as interval notation but provides an immediate visual understanding of the solution.

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Most popular questions from this chapter

Solve each inequality. Write the solution set in interval notation and graph it. $$ \frac{3 a-4}{-5}<\frac{3 a+15}{-5} $$

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