Math Problem Statement
Directions: A) write each interval as an inequality B) then graph the solutions C) show ALL work and show ALL of the numbers on the numberline
- (-infinity symbol,2]
- (-infinity symbol,-1] or (3,infinity symbol)
Solution
Let's go through each interval step by step.
Problem 10:
A) Write as an inequality:
This interval represents all real numbers less than or equal to . In inequality form:
B) Graph the solution:
- The interval is from negative infinity up to 2, including 2 itself.
- On the number line:
- Use a filled circle (•) at because it includes 2.
- The arrow going left from 2 indicates the negative direction (towards negative infinity).
Graphically, it would look like this:
<====•----------------->
2
C) Show ALL work:
The interval means that can be any number less than or equal to 2. This is directly written as the inequality . The number line shows that all values from negative infinity up to 2 are valid, and the circle at is filled to show the inclusion of .
Problem 11:
A) Write as an inequality:
This is a union of two intervals:
- means .
- means .
So, the inequality for this is:
B) Graph the solution:
- The interval includes all numbers less than or equal to , so we place a filled circle at and draw a line going to the left.
- The interval includes all numbers greater than , so we place an open circle at and draw a line going to the right.
Graphically:
<====• ( )====>
-1 3
C) Show ALL work:
The solution consists of two parts:
- corresponds to the interval .
- corresponds to the interval .
On the number line, we use a filled circle at to show inclusion and an open circle at to show exclusion, with the arrows pointing left and right for the respective ranges.
Further Learning Opportunities:
- How does the graph differ for open vs. closed intervals?
- What happens if we change the inequality to ?
- How do you solve compound inequalities like this using algebra?
- How can interval notation be used to represent solutions for quadratic inequalities?
- How would you solve a similar problem with more than two intervals?
Tip:
When writing interval notation, always remember that:
- Square brackets [ ] denote inclusion (≤ or ≥).
- Parentheses ( ) denote exclusion (< or >).
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Inequalities
Interval Notation
Graphing
Set Theory
Formulas
x ≤ 2
x ≤ -1 or x > 3
Theorems
Interval Notation Theorem
Inequality Graphing Theorem
Suitable Grade Level
Grades 7-9