Problem 11
Question
In Exercises 1-12, graph the solutions of each inequality on a number line.
$$-1
Step-by-Step Solution
Verified Answer
The graph of the solution to the inequality \(-1
1Step 1: Identify the inequality
Here, the inequality given is \(-1
2Step 2: Set up the number line
We will draw a number line and mark the intervals. It's usually best to mark off the numbers mentioned in the equation and a few to both sides of them. -1 and 4 should be marked, and pick numbers less than -1 and greater than 4 to give context.
3Step 3: Graph the solution
The solution of the inequality \(-1
Key Concepts
Number LineInequality SolutionsOpen Circle Notation
Number Line
Understanding the number line is crucial when dealing with inequalities. A number line is a straight, horizontal line that has points which correspond to numbers. The numbers are spaced equally along the line, and it extends infinitely in both the positive and negative directions, much like a real-world ruler that never ends.
When you're graphing inequalities on a number line, you begin by marking the boundary numbers from the inequality—these are your reference points. In the given exercise, -1 and 4 are these reference points. The space in between the boundary numbers represents all the possible solutions to the inequality.
Inequalities often require you to think about ranges of numbers rather than just specific values. By using a number line, you can visually represent and reason about these ranges. For students, remembering to evenly space numbers and correctly position negative and positive values is essential.
When you're graphing inequalities on a number line, you begin by marking the boundary numbers from the inequality—these are your reference points. In the given exercise, -1 and 4 are these reference points. The space in between the boundary numbers represents all the possible solutions to the inequality.
Inequalities often require you to think about ranges of numbers rather than just specific values. By using a number line, you can visually represent and reason about these ranges. For students, remembering to evenly space numbers and correctly position negative and positive values is essential.
Inequality Solutions
An inequality solution represents a set of numbers that satisfy the given inequality. Unlike equations where you might find a single answer, inequalities often have a range of possible values that 'work'.
In the case of the inequality \( -1 < x < 4 \) from the exercise, the solutions are all the real numbers that are greater than -1 and less than 4. This doesn't just include whole numbers or integers; fractions and decimals within this interval also belong to the solution set.
It's worth noting that solving the inequality involves understanding whether the boundary numbers are included or not. In our exercise, the inequality uses '<' (less than), not '≤' (less than or equal to), which means -1 and 4 are not included in the solutions. A student must recognize these symbols and what they signify about the inclusion of boundary numbers in the solution set.
In the case of the inequality \( -1 < x < 4 \) from the exercise, the solutions are all the real numbers that are greater than -1 and less than 4. This doesn't just include whole numbers or integers; fractions and decimals within this interval also belong to the solution set.
It's worth noting that solving the inequality involves understanding whether the boundary numbers are included or not. In our exercise, the inequality uses '<' (less than), not '≤' (less than or equal to), which means -1 and 4 are not included in the solutions. A student must recognize these symbols and what they signify about the inclusion of boundary numbers in the solution set.
Open Circle Notation
Open circle notation is a visual tool used on a number line to indicate that a particular value is not included in the set of solutions for an inequality. When graphing, you'll place a small open circle over the number to show that while the value is a reference point, it's not actually part of the solution set.
In our exercise, where the inequality is \( -1 < x < 4 \) , there would be open circles above -1 and 4. This is because the value of \(x\) is strictly greater than -1 and strictly less than 4, so while the number line will have an unbroken line between these two points indicating all the values in between are included, the endpoints themselves are excluded.
The concept of open circle notation highlights the difference between 'strict' inequalities ( < and >) and 'inclusive' inequalities ('≤' and '≥'). If the inequality had included the endpoints, we would use closed circles instead to show that these values are included. For students practicing inequalities, mastering this notation is key to accurately conveying solutions.
In our exercise, where the inequality is \( -1 < x < 4 \) , there would be open circles above -1 and 4. This is because the value of \(x\) is strictly greater than -1 and strictly less than 4, so while the number line will have an unbroken line between these two points indicating all the values in between are included, the endpoints themselves are excluded.
The concept of open circle notation highlights the difference between 'strict' inequalities ( < and >) and 'inclusive' inequalities ('≤' and '≥'). If the inequality had included the endpoints, we would use closed circles instead to show that these values are included. For students practicing inequalities, mastering this notation is key to accurately conveying solutions.
Other exercises in this chapter
Problem 11
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