About Me
Math for Grade 7
1 Number Sense and Operations
1-1 Integers
1-1 1 Understanding positive and negative numbers
1-1 2 Comparing and ordering integers
1-1 3 Absolute value
1-1 4 Adding and subtracting integers
1-1 5 Multiplying and dividing integers
1-2 Rational Numbers
1-2 1 Understanding fractions, decimals, and mixed numbers
1-2 2 Comparing and ordering rational numbers
1-2 3 Converting between fractions, decimals, and percents
1-2 4 Adding and subtracting fractions and mixed numbers
1-2 5 Multiplying and dividing fractions and mixed numbers
1-3 Exponents and Roots
1-3 1 Understanding exponents
1-3 2 Laws of exponents
1-3 3 Square roots and cube roots
2 Algebra
2-1 Expressions and Equations
2-1 1 Writing algebraic expressions
2-1 2 Evaluating algebraic expressions
2-1 3 Solving one-step and two-step equations
2-1 4 Solving inequalities
2-2 Patterns and Functions
2-2 1 Identifying and extending patterns
2-2 2 Representing patterns with tables, graphs, and equations
2-2 3 Understanding functions and function notation
3 Geometry
3-1 Shapes and Angles
3-1 1 Classifying polygons
3-1 2 Measuring and drawing angles
3-1 3 Understanding complementary and supplementary angles
3-2 Transformations
3-2 1 Understanding translations, reflections, and rotations
3-2 2 Identifying congruent and similar figures
3-3 Perimeter, Area, and Volume
3-3 1 Calculating perimeter and area of polygons
3-3 2 Understanding and calculating the area of circles
3-3 3 Calculating the volume of rectangular prisms
4 Data and Probability
4-1 Data Representation
4-1 1 Collecting and organizing data
4-1 2 Creating and interpreting bar graphs, line graphs, and pie charts
4-1 3 Understanding mean, median, mode, and range
4-2 Probability
4-2 1 Understanding probability as a ratio
4-2 2 Calculating simple probabilities
4-2 3 Understanding experimental versus theoretical probability
5 Problem Solving and Critical Thinking
5-1 Strategies for Problem Solving
5-1 1 Using logical reasoning and critical thinking
5-1 2 Applying the problem-solving process
5-1 3 Using estimation and approximation
5-2 Real-World Applications
5-2 1 Applying mathematical concepts to real-world scenarios
5-2 2 Understanding the relevance of mathematics in daily life
Solving Inequalities

Solving Inequalities

Key Concepts

1. **Inequalities**: Mathematical statements that show the relationship between two expressions that are not equal.

2. **Symbols**: \(<\) (less than), \(>\) (greater than), \(\leq\) (less than or equal to), \(\geq\) (greater than or equal to).

3. **Properties**: Similar to equations, but with special rules when multiplying or dividing by a negative number.

Detailed Explanation

Inequalities

Inequalities are used to compare two expressions. For example, \(3x + 2 > 7\) is an inequality. The goal is to find the values of \(x\) that make the statement true.

Symbols

The symbols \(<\) and \(>\) indicate strict inequalities, while \(\leq\) and \(\geq\) indicate inequalities that include equality. For example, \(x \leq 5\) means \(x\) can be 5 or any number less than 5.

Properties

When solving inequalities, you can add or subtract the same number from both sides without changing the inequality. However, when multiplying or dividing by a negative number, you must reverse the inequality symbol. For example, if \(-2x > 4\), dividing by \(-2\) gives \(x < -2\).

Examples

Example 1: Solve \(2x + 3 > 7\).

Solution: Subtract 3 from both sides: \(2x > 4\).

Divide both sides by 2: \(x > 2\).

Example 2: Solve \(-3x + 5 \leq 11\).

Solution: Subtract 5 from both sides: \(-3x \leq 6\).

Divide both sides by \(-3\) and reverse the inequality: \(x \geq -2\).

Example 3: Solve \(4x - 2 < 10\).

Solution: Add 2 to both sides: \(4x < 12\).

Divide both sides by 4: \(x < 3\).

Analogies

Think of inequalities as a balance scale. When you add or subtract the same weight from both sides, the scale remains balanced. However, if you multiply or divide by a negative number, it's like flipping the scale, causing the heavier side to become the lighter side.

Practical Application

Solving inequalities is crucial in various real-world scenarios, such as budgeting, determining ranges of values in data analysis, and setting constraints in engineering and physics problems. By mastering these concepts, you can better analyze and solve problems in everyday life.

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