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Chapter 8: Problem 27

Solve each inequality. Then graph the solution on a number line. $$-3+z>2$$

Short Answer

Expert verified
The solution is \(z > 5\). Graph it by shading the number line to the right of 5 with an open circle on 5.

Step by step solution

01

Isolate the Variable

To solve the inequality \(-3 + z > 2\), we need to isolate \(z\) on one side of the inequality. Start by adding 3 to both sides to undo the subtraction of 3 from \(z\). This will give us: \[-3 + z + 3 > 2 + 3\] Simplify to get: \[z > 5\]
02

Graph the Solution

With the inequality \(z > 5\), identify the portion of the number line where \(z\) is greater than 5. This means we need to shade the region to the right of 5. Since \(z\) can be any number greater than 5, place an open circle at 5 to indicate that 5 itself is not included in the solution set.

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Key Concepts

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

Number Line Graphing for Inequalities
When solving inequalities, graphing the solution on a number line can help visualize the range of solutions. In our exercise, after solving the inequality \(z > 5\), we need to represent this visually. Number line graphing presents the set of values that a variable, such as \(z\), can take. For \(z > 5\), you would:
  • Draw a horizontal line which represents the number line, marking numbers clearly at regular intervals.
  • Locate the number 5 on this line.
  • Use an open circle at 5 to show that 5 is not included in the solutions.
  • Shade or draw an arrow extending to the right to indicate all numbers greater than 5. This illustrates that the solution includes numbers just larger than 5, continuing indefinitely.
Graphing ensures you understand both the specific and infinite nature of solutions in inequalities. It emphasizes which numbers satisfy the inequality relationship.
Isolate the Variable: A Key Concept
Isolating the variable is a critical step in solving inequalities. It means getting the unknown variable, in this case \(z\), alone on one side of the inequality. By doing so, you can clearly identify its range of possible values. In the inequality \(-3 + z > 2\),
  • Start by removing anything attached to the variable by adding or subtracting. For our inequality, add 3 to both sides to reverse the subtraction done to \(z\).
  • This step changes \(-3 + z > 2\) into \(z > 5\), simplifying it to an inequality that's easy to interpret.
  • Performing such operations keeps the inequality balanced. Always do the same operation on both sides to keep it valid.
Isolating ensures a clear view of the variable's potential values, setting the stage for solving the inequality.
Understanding Inequality Solutions
Inequalities express a range of solutions, rather than just one fixed answer. When we solved \(z > 5\), it was not about finding a single answer, but expressing a relationship where \(z\) can be any number greater than 5. Key points about inequality solutions include:
  • Inequalities use symbols like \(>\), \(<\), \(\geq\), and \(\leq\) to show relationships between values.
  • Solutions to inequalities are often infinite sets of numbers. For example, \(z > 5\) means any number, no matter how large, is a solution if it is greater than 5.
  • Open and closed circles on a graph help indicate if boundary numbers are included. For \(>\) or \(<\), use an open circle; and for \(\geq\) or \(\leq\), use a closed circle.
An inequality solution highlights possible values, rather than a set value, enhancing our understanding of numerical relationships.

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Most popular questions from this chapter

PREREQUISITE SKILL Solve each equation. $$\frac{c}{-4}=12$$

Solve each inequality and check your solution. Then graph the solution on a number line. $$12 a-4>20$$

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Find each quotient. Write in simplest form. $$\frac{3}{2} \div \frac{1}{5}$$
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