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

  1. (-infinity symbol,2]
  2. (-infinity symbol,-1] or (3,infinity symbol)

Solution

Let's go through each interval step by step.

Problem 10: (,2](- \infty, 2]

A) Write as an inequality:

This interval represents all real numbers less than or equal to 22. In inequality form:

x2x \leq 2

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 x=2x = 2 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 (,2](- \infty, 2] means that xx can be any number less than or equal to 2. This is directly written as the inequality x2x \leq 2. The number line shows that all values from negative infinity up to 2 are valid, and the circle at 22 is filled to show the inclusion of 22.


Problem 11: (,1](3,)(- \infty, -1] \cup (3, \infty)

A) Write as an inequality:

This is a union of two intervals:

  1. (,1](- \infty, -1] means x1x \leq -1.
  2. (3,)(3, \infty) means x>3x > 3.

So, the inequality for this is:

x1orx>3x \leq -1 \quad \text{or} \quad x > 3

B) Graph the solution:

  • The interval (,1](- \infty, -1] includes all numbers less than or equal to 1-1, so we place a filled circle at 1-1 and draw a line going to the left.
  • The interval (3,)(3, \infty) includes all numbers greater than 33, so we place an open circle at 33 and draw a line going to the right.

Graphically:

<====•                     ( )====>
       -1                     3

C) Show ALL work:

The solution consists of two parts:

  1. x1x \leq -1 corresponds to the interval (,1](- \infty, -1].
  2. x>3x > 3 corresponds to the interval (3,)(3, \infty).

On the number line, we use a filled circle at 1-1 to show inclusion and an open circle at 33 to show exclusion, with the arrows pointing left and right for the respective ranges.


Further Learning Opportunities:

  1. How does the graph differ for open vs. closed intervals?
  2. What happens if we change the inequality to x1x \geq -1?
  3. How do you solve compound inequalities like this using algebra?
  4. How can interval notation be used to represent solutions for quadratic inequalities?
  5. 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 >).

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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