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Chapter 8: Problem 8

Solve each inequality. Then graph the solution on a number line. $$x-6 \leq 4$$

Short Answer

Expert verified
The solution is \( x \leq 10 \), and it is represented by a closed dot on 10 with a line extending left on a number line.

Step by step solution

01

Identify the Inequality

The given inequality is \( x - 6 \leq 4 \). We need to solve for \( x \) by isolating it on one side of the inequality.
02

Add 6 to Both Sides

To isolate \( x \), add 6 to both sides of the inequality: \( x - 6 + 6 \leq 4 + 6 \). This simplifies to \( x \leq 10 \).
03

Interpret the Solution

The solution of the inequality is \( x \leq 10 \), which means \( x \) can be any number less than or equal to 10.
04

Graph the Solution on a Number Line

To graph \( x \leq 10 \), draw a number line and place a closed dot on 10, indicating that 10 is included in the solution. Draw a line extending to the left to show all numbers less than 10 are 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.

Inequality Graphing
Graphing an inequality is a visual representation of all possible solutions to the inequality. It helps you understand which values satisfy the condition described. To graph the inequality \( x \leq 10 \):
  • First, identify the endpoint which is 10 in this case.
  • You want to show that all values less than or equal to 10 satisfy this inequality.
  • Use a closed dot to mark the 10 on the number line, signifying it's part of the solution.
  • Draw a line extending to the left from 10 to indicate that all numbers smaller than 10 are part of the solution set.
Learning how to graph inequalities is an important step in understanding their solutions. It transforms abstract math into tangible visual information.
Number Line
A number line is a horizontal straight line used to visualize numbers and their relationships. It acts as a simple and effective tool when dealing with inequalities. On a number line, each point corresponds to a real number. Here鈥檚 how you can effectively use it:
  • Numbers to the right are greater than those to the left, aligning with our natural understanding of order.
  • Important in graphing inequalities; it shows which numbers meet the condition set by the inequality.
  • Draw the number line with enough space between numbers to mark points clearly.
  • Select an appropriate range that extends beyond the key points to visualize the whole solution.
Understanding the number line is key to effectively graphing and interpreting inequalities.
Isolating Variables
Isolating the variable is a fundamental step in solving inequalities, just like solving equations. It's about getting the variable by itself on one side. Here鈥檚 how you can isolate a variable step by step:
  • Start by identifying which terms need to be moved to isolate the variable. For \( x - 6 \leq 4 \), this means moving the \(-6\).
  • To undo subtraction, add the same number to both sides \( x - 6 + 6 \leq 4 + 6 \).
  • Simplify both sides to achieve the isolated variable: \( x \leq 10 \).
Isolating variables helps simplify the problem and clearly shows all the numbers the variable could represent.
Interpreting Solutions
Interpreting the solution of an inequality involves understanding what the solution set means. With \( x \leq 10 \), interpreting involves:
  • Recognizing \( x \leq 10 \) includes every number that is 10 or below.
  • Visualizing these numbers on the number line helps easily see the range of solutions.
  • Realizing that this interpretation affects how you use or apply the inequality's solution in real-world situations, such as measuring limits or constraints.
Taking the time to accurately interpret solutions ensures you fully grasp which numbers fulfill the inequality requirements. It helps apply mathematical logic to practical problems.

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