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Chapter 6: Problem 29

Graph the inequality. \(-10 \leq-x\)

Short Answer

Expert verified
The graph should have a filled point at 10 and the line should extend to the left, representing all the numbers which are less than or equal to 10.

Step by step solution

01

Simplify the Inequality

First, -x should be transformed into a positive form. That can be achieved by multiplying both sides of the inequality by -1. However, when multiplying or dividing both sides of an inequality by a negative number, the inequality symbol must be reversed. The simplified inequality now is \(x \leq 10\).
02

Identify the Number Line

The inequality represents a collection of numbers which are less than or equal to 10. These values will be represented on the number line. Hence, every value to the left of 10, including 10, will be included in the solution of the inequality.
03

Graph the Inequality

Plot the point at x = 10 and fill the dot since the solution includes 10 (as indicated by the \(\leq\)). Draw a line to the left of 10, indicating all numbers less than 10 are part of the solution.

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Key Concepts

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

Solving Inequalities
Solving inequalities is much like solving equations, but with a few key differences. Inequalities use symbols like \(<, >, \leq, \geq\) to show that values are either greater, less than or equal. When you solve an inequality, you're finding a range of numbers that make the inequality true.
To solve them:
  • Perform operations like addition, subtraction, multiplication, or division on both sides, similar to an equation.
  • Be cautious! If you multiply or divide by a negative number, flip the inequality symbol. This is crucial and often a common mistake.
For example, in solving \(-10 \leq -x\), we multiply both sides by \(-1\) and flip the inequality sign to get \(x \leq 10\).
Keep these rules in mind and practice to improve your skills in solving a variety of inequalities.
Number Line
A number line is a valuable tool for representing solutions of inequalities visually. It is a straight line where each point corresponds to a number.
To graph an inequality:
  • Identify the key number from the inequality, such as \(x \leq 10\).
  • Plot this number on the number line. Use a filled dot if the number is included (\(\leq\) or \(\geq\)). Use an open dot if it's not included (\(<\) or \(>\)).
  • Draw an arrow or a line starting from the dot, moving in the direction of the inequality. For \(x \leq 10\), draw to the left to include values less than or equal to 10.
This creates a simple way of visualizing solutions, making it easier to interpret the range of values that satisfy the inequality.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operation symbols. When working with inequalities, understanding how to manipulate these expressions is crucial.
In an inequality like \(-10 \leq -x\), the expression \(-x\) includes a variable (\(x\)) and a coefficient (-1). Here鈥檚 how you can work with them:
  • Isolate the variable on one side of the inequality, much like solving an equation.
  • Simplify each side by combining like terms or distributing as needed.
  • Remember to reverse the inequality when multiplying or dividing by a negative number.
Mastering algebraic expressions helps in effectively understanding and solving inequalities, allowing for more complex problems to be tackled with confidence.

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