Math Problem Statement

Solve various types of inequalities from a worksheet that includes linear, quadratic, absolute value, compound, and rational inequalities.

Solution

I see that you've uploaded an image related to solving inequalities. The image contains a worksheet with inequality problems and explanations about how to solve absolute value inequalities. Here's a breakdown of the contents:

  1. Left Panel: It shows the steps for solving an inequality of the form 3x1263x - 12 \geq 6 and illustrates how to split and solve inequalities. It includes number line representations for the solution range.

  2. Middle Panel: This contains a list of inequalities to solve, categorized into different types such as:

    • Linear inequalities like 2x4<13x2x - 4 < 1 - 3x
    • Quadratic inequalities such as x236x^2 \leq 36
    • Absolute value inequalities like x<5|x| < 5
    • Compound inequalities like 2<x1<62 < x - 1 < 6
    • Rational inequalities like 1x+1<0\frac{1}{x+1} < 0
  3. Right Panel: This discusses solving absolute value inequalities of the form ax+bc|ax + b| \geq c. It explains the process of splitting the inequality into two cases and solving each case separately.

Would you like help solving any specific inequality from the worksheet or more details on how to solve absolute value inequalities?

Here are 5 related questions for further practice:

  1. How do you solve the inequality 2x4<13x2x - 4 < 1 - 3x from 1.1?
  2. Can you explain how to solve the quadratic inequality x23x10>0x^2 - 3x - 10 > 0 in 2.2?
  3. What is the solution to the absolute value inequality x5|x| \leq 5 in 4.1?
  4. How do you approach compound inequalities like 2<x1<62 < x - 1 < 6 in 5.1?
  5. What method is used to solve rational inequalities like x+32x+10\frac{x + 3}{2x + 1} \leq 0 in 3.2?

Tip: When solving inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number!

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Math Problem Analysis

Mathematical Concepts

Linear Inequalities
Quadratic Inequalities
Absolute Value Inequalities
Compound Inequalities
Rational Inequalities

Formulas

ax + b ≥ c
Quadratic inequality: ax^2 + bx + c > 0
Absolute value inequality: |ax + b| ≥ c
Compound inequality: a < x < b

Theorems

Properties of inequalities
Quadratic formula
Absolute value inequality theorem

Suitable Grade Level

Grades 8-11