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Chapter 7: Problem 10

Graph inequality. \(x \leq-3\)

Short Answer

Expert verified
To graph the inequality \(x \leq -3\), draw a number line with a closed circle at -3, indicating that x can be -3, and then extend a line to its left, indicating that x can take any value less than -3 too.

Step by step solution

01

Understanding inequality

In the expression, \(x \leq -3\), the \(\leq\) sign means 'less than or equal to'. This means we are looking for all x values that are less than or equal to -3.
02

Plotting the graph

Begin graphing by drawing a number line. The number line should include -3 since it's the boundary point. Since the inequality is \(\leq\), place a closed circle at -3 as it includes -3. The closed circle signifies that the value -3 is included in the solution.
03

Representing inequalities on the number line

Next, since the inequality signifies \(x\) values that are 'less than or equal to', we will extend a line from the point -3 towards the lesser numbers or to the left on the number line. All the values towards left of -3 on the number line represent the range of \(x\) that satisfies the inequality \(x \leq -3\).

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Key Concepts

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

Number Line
In mathematics, a number line is a visual representation of numbers placed at regular intervals along a straight line. It serves as a powerful tool for understanding various mathematical concepts, including inequalities, as seen in the exercise given.
The number line is continuous and infinite in both directions, containing both positive and negative numbers, zero, and fractions or decimals when necessary. It helps us visualize numbers and their relative positions to each other.
When graphing inequalities like \(x \leq -3\), the number line helps us demonstrate which numbers satisfy the condition by showing a range of values.
  • Numbers on the line to the left represent lesser values, while those to the right are greater.
  • By marking specific points, such as -3 in our exercise, we can easily identify included or excluded values based on the inequality's requirements.
  • The use of a number line simplifies understanding, especially when dealing with inequalities involving limitations or boundaries.
Closed Circle
A closed circle is a vital concept when graphing inequalities, as it indicates that the endpoint is included in the solution set. In our exercise, the inequality \(x \leq -3\) means -3 is part of the solution.
This is crucial because, without a closed circle, the understanding of whether the boundary value is included or not could be ambiguous.
  • Closed circle: Used for inequalities that have 'less than or equal to' (\(\leq\)) or 'greater than or equal to' (\(\geq\)) symbols.
  • A closed circle is drawn at the point -3 on the number line, signifying that -3 is a valid solution to the inequality.
  • Visually, it appears as a fully shaded circle positioned directly on the boundary point, ensuring no confusion about inclusion.
Understanding and correctly using a closed circle ensures accuracy when presenting solutions on a number line.
Less Than or Equal to Inequality
The inequality \(x \leq -3\) uses the symbol \(\leq\), which stands for 'less than or equal to.' It incorporates two components: values less than -3 and the value -3 itself.
This dual nature makes it integral to understand how this type of inequality functions and how it impacts graphical representation.
  • The sign indicates solutions can be any number that is either smaller or exactly equal to the boundary point.
  • Graphing it: On a number line, a closed circle at -3, with a line extending leftward, reflects the inequality.
  • This extension shows all values that meet the condition \(x \leq -3\), emphasizing inclusivity of the number -3.
Using less than or equal to inequalities provides a precise method for expressing ranges of numbers, essential for accurately solving and interpreting real-world problems.

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Most popular questions from this chapter

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities. $$y \geq x^{2}-4$$

Let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The sum of two numbers is 10 and their product is \(24 .\) Find the numbers.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Every linear system has infinitely many ordered-pair solutions.

Explain how to solve a system of equations using the substitution method. Use \(y=3-3 x\) and \(3 x+4 y=6\) to illustrate your explanation.

The group should write four different word problems that can be solved using a system of linear equations in two variables. All of the problems should be on different topics. The group should turn in the four problems and their algebraic solutions.
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