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Inequalities are similar to equations but instead of an equal sign, they show a relationship where one side is greater or less than the other. When solving inequalities, our goal is to isolate the variable on one side while following specific rules.

For example, when multiplying or dividing both sides of an inequality by a negative number, we must reverse the inequality sign. This is crucial to obtaining the correct solution. Consider the inequality -7x ≥ -7. When divided by -7, the inequality sign flips from '≥' to '≤,' resulting in x ≤ 1.

Compound inequalities involve multiple inequalities combined by the words 'and' or 'or.' The solution for 'and' inequalities must satisfy both conditions, while 'or' inequalities only need to meet one condition. Let's look at our example: 2-7x ≥ -5 or 4(x-3) + 7 > 3. We solve each part separately and then combine their solutions.

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Understanding algebraic expressions is a fundamental aspect of solving inequalities.

Let's consider the expression 4(x-3) + 7 > 3. Here, we start by distributing the 4 to both terms inside the parentheses: 4 * x and 4 * -3, resulting in 4x - 12. Adding 7 to this gives us the simplified expression 4x - 5 > 3. Breaking down complex expressions into simpler parts helps in clearly understanding each step.

Simplification involves combining like terms and following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). By mastering the manipulation of algebraic expressions, solving inequalities becomes a more straightforward task.

A step-by-step approach makes solving complex problems manageable. Let's break down the given exercise into simpler steps:

• Solve the first inequality: 2 - 7x ≥ -5. Subtract 2 from both sides: -7x ≥ -7. Then divide by -7, flipping the sign: x ≤ 1.
• Solve the second inequality: 4(x-3) + 7 > 3. Distribute the 4: 4x - 12 + 7 > 3. Simplify: 4x - 5 > 3. Add 5 to both sides: 4x > 8. Finally, divide by 4: x > 2.
• Combine the solutions: Since the original problem states 'or,' the final result is x ≤ 1 or x > 2.

Detailed, step-by-step solutions not only help in understanding but also ensuring no steps or rules are overlooked. This approach highlights the logical sequence required to solve inequalities efficiently, boosting both understanding and accuracy.