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Linear inequalities are similar to linear equations but with inequality signs like <, >, ≤, or ≥ instead of an equal sign. When solving them, our goal is to find the range of values for the variable that makes the inequality true.

For example, consider the inequality \(-3x - 2 < 7\). This tells us that for some values of \x\, the expression \(-3x - 2\) is less than 7. To find those values, we follow a series of algebraic steps, ensuring that we maintain the inequality's direction unless specific rules apply (more on this later!).

Linear inequalities can be graphed on a number line, showing all possible solutions. When the sign is < or >, the range will be shown with an open circle (not including the endpoint). Signs ≤ or ≥ use a closed circle, including the endpoint.

An important rule while solving inequalities is recognizing when and why to reverse the inequality sign. This occurs specifically when multiplying or dividing both sides of the inequality by a negative number. Here's why:

If you consider the numbers 2 and 3, it’s clear that 2 < 3. However, if you multiply both by -1, you get -2 and -3. Now, -2 is not less than -3; in fact, -2 > -3. This demonstrates why the inequality sign needs to be flipped.

In our example, \(-3x < 9\), when we divide both sides by -3, the inequality signs flips, leading to \x > -3\.

Quick tips for remembering:

• Multiplying or dividing by a positive number: Keep the inequality direction the same.
• Multiplying or dividing by a negative number: Flip the inequality sign.

This simple rule keeps the solution set accurate and ensures the inequality remains valid.

Algebraic manipulation refers to the steps we take to isolate the variable and solve the inequality. For our inequality \(-3x - 2 < 7\), we follow these steps:

• Add or Subtract: Start by eliminating any constants from the variable side. Here, we add 2 to each side: \(-3x - 2 + 2 < 7 + 2\), simplifying to \(-3x < 9\).
• Multiply or Divide: Next, divide or multiply to solve for the variable. Since \-3\ is multiplied with \x\, we divide both sides by -3 (and flip the inequality sign): \(\frac{-3x}{-3} > \frac{9}{-3}\), simplifying to \x > -3\.

These steps break down to simple operations that isolate \x\, showing that the solutions to the inequality include all numbers greater than -3.

Understanding and mastering these manipulations will allow you to confidently solve inequalities and ensure you're applying each step correctly, leading to accurate results.