Math Problem Statement

Solve the inequality. Graph the solutions. StartAbsoluteValue x EndAbsoluteValueplus2greater than10

Solution

The inequality is:

x+2>10|x| + 2 > 10

Step 1: Isolate the absolute value term.

We start by subtracting 2 from both sides of the inequality:

x>102|x| > 10 - 2

x>8|x| > 8

Step 2: Break the absolute value inequality into two cases.

Since x>8|x| > 8, the solutions will be when:

x>8orx<8x > 8 \quad \text{or} \quad x < -8

Step 3: Graph the solution.

The solution consists of all values of xx greater than 8 and less than -8. These are two open intervals:

x(,8)(8,)x \in (-\infty, -8) \cup (8, \infty)

On a number line, this would be represented by two rays extending to the left from 8-8 and to the right from 88, with open circles at 8-8 and 88, indicating that these points are not included in the solution.

Graph Description:

  1. A number line with two open circles at 8-8 and 88.
  2. Arrows extending left from 8-8 and right from 88, representing all values less than 8-8 and greater than 88.

Would you like me to generate this graph for you?

Relative Questions:

  1. How do absolute value inequalities differ from regular inequalities?
  2. What happens when we reverse the inequality symbol (e.g., x+2<10|x| + 2 < 10)?
  3. How would the solution change if the inequality was x+210|x| + 2 \geq 10?
  4. Can you solve a similar inequality, such as x5>7|x - 5| > 7?
  5. How do we graph compound inequalities on a number line?

Tip: Always isolate the absolute value term first when solving inequalities!

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value Inequalities
Inequalities

Formulas

|x| > a when x > a or x < -a

Theorems

Absolute Value Theorem

Suitable Grade Level

Grades 8-10