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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Solving Inequalities</title>
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<body>
<header>
<h1>Solving Inequalities</h1>
</header>
<div class="content">
<h2>Solving Inequalities Deeply</h2>
<p>An inequality is a mathematical expression that relates two values that may not be equal. It is similar to an equation, but instead of using an equal sign (<code>=</code>), we use inequality symbols to show that one side of the expression is larger or smaller than the other side.</p>
<h3>1. Types of Inequalities</h3>
<p>Inequalities are divided into different types based on the symbol used:</p>
<ul>
<li><strong>Strict inequalities</strong>:
<ul>
<li><code><</code> (less than)</li>
<li><code>></code> (greater than)</li>
</ul>
</li>
<li><strong>Non-strict inequalities</strong>:
<ul>
<li><code>≤</code> (less than or equal to)</li>
<li><code>≥</code> (greater than or equal to)</li>
</ul>
</li>
</ul>
<p>Each of these symbols represents a range of values that satisfy the inequality.</p>
<h3>2. Solving Linear Inequalities</h3>
<p>A linear inequality involves a linear expression, i.e., the highest degree of the variable is 1. The basic rules for solving linear inequalities are similar to solving equations, but we must take special care when multiplying or dividing by negative numbers.</p>
<h3>Steps for Solving Linear Inequalities</h3>
<ol>
<li>Isolate the variable on one side of the inequality (similar to solving an equation).</li>
<li>Simplify both sides if needed (combine like terms).</li>
<li>If you multiply or divide by a negative number, you must reverse the inequality sign.</li>
<li>Graph the solution on a number line (if required).</li>
</ol>
<h3>Example 1:</h3>
<p>Solve: <code>2x - 3 ≥ 7</code></p>
<p><strong>Step 1</strong>: Add 3 to both sides:</p>
<p><code>2x ≥ 10</code></p>
<p><strong>Step 2</strong>: Divide both sides by 2:</p>
<p><code>x ≥ 5</code></p>
<p>The solution is <code>x ≥ 5</code>.</p>
<h3>Example 2:</h3>
<p>Solve: <code>-3x < 9</code></p>
<p><strong>Step 1</strong>: Divide both sides by -3 (remember to reverse the inequality sign):</p>
<p><code>x > -3</code></p>
<p>The solution is <code>x > -3</code>.</p>
<h3>3. Solving Compound Inequalities</h3>
<p>Compound inequalities involve two or more inequalities that are joined together by the words "and" or "or". The solution is found by solving each inequality separately and then combining the results.</p>
<h3>"And" Compound Inequality:</h3>
<p>Example: Solve <code>3 ≤ x + 2 ≤ 5</code></p>
<p><strong>Step 1</strong>: Solve the left inequality: <code>3 ≤ x + 2</code> ⟶ <code>x ≥ 1</code></p>
<p><strong>Step 2</strong>: Solve the right inequality: <code>x + 2 ≤ 5</code> ⟶ <code>x ≤ 3</code></p>
<p><strong>Step 3</strong>: Combine both parts: The solution is <code>1 ≤ x ≤ 3</code>.</p>
<h3>"Or" Compound Inequality:</h3>
<p>Example: Solve <code>x < -2 or x ≥ 4</code></p>
<p>The solution is the union of both inequalities: <code>x < -2 or x ≥ 4</code>.</p>
<h3>4. Solving Absolute Value Inequalities</h3>
<p>Absolute value inequalities involve expressions that have absolute value bars around the variable or expression. The absolute value represents the distance of a number from zero on the number line, regardless of direction.</p>
<p><strong>For inequalities of the form</strong> <code>|x| < a</code>:</p>
<p>The solution is <code>-a < x < a</code>.</p>
<p><strong>For inequalities of the form</strong> <code>|x| > a</code>:</p>
<p>The solution is <code>x < -a</code> or <code>x > a</code>.</p>
<h3>Example 1:</h3>
<p>Solve: <code>|x - 3| < 5</code></p>
<p><code>-5 < x - 3 < 5</code></p>
<p>Add 3 to all parts: <code>-2 < x < 8</code></p>
<p>The solution is <code>-2 < x < 8</code>.</p>
<h3>Example 2:</h3>
<p>Solve: <code>|x + 4| ≥ 7</code></p>
<p>This gives two inequalities:</p>
<ul>
<li><code>x + 4 ≥ 7</code> ⟶ <code>x ≥ 3</code></li>
<li><code>x + 4 ≤ -7</code> ⟶ <code>x ≤ -11</code></li>
</ul>
<p>The solution is <code>x ≤ -11 or x ≥ 3</code>.</p>
<h3>5. Graphing Inequalities</h3>
<p>When graphing inequalities on a number line:</p>
<ul>
<li>For strict inequalities (e.g., <code>x > 3</code>), use an open circle at the point.</li>
<li>For non-strict inequalities (e.g., <code>x ≥ 3</code>), use a closed circle at the point.</li>
<li>Use arrows to indicate the direction of the solution (to the right for greater values and to the left for smaller values).</li>
</ul>
<h3>Formulas Related to Solving Inequalities</h3>
<ul>
<li><strong>Addition and Subtraction Rule:</strong> If <code>a ≤ b</code>, then <code>a + c ≤ b + c</code> and <code>a - c ≤ b - c</code>, for any real number <code>c</code>.</li>
<li><strong>Multiplication and Division Rule (by a positive number):</strong> If <code>a ≤ b</code>, then <code>a × c ≤ b × c</code> and <code>a ÷ c ≤ b ÷ c</code>, for any positive real number <code>c</code>.</li>
<li><strong>Multiplication and Division Rule (by a negative number):</strong> If <code>a ≤ b</code>, then <code>a × c ≥ b × c</code> and <code>a ÷ c ≥ b ÷ c</code>, for any negative real number <code>c</code> (note: the inequality flips when multiplying or dividing by negative numbers).</li>
</ul>
<h3>6. Special Tips for Solving Inequalities:</h3>
<ul>
<li>Always be careful when multiplying or dividing by negative numbers, as the inequality sign will flip.</li>
<li>Graph the solution on a number line for a visual representation.</li>
<li>Check the boundary points (if any) to ensure they satisfy the inequality.</li>
</ul>
</div>
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<p>© 2025 Yoseph Feyisa Wegi | All Rights Reserved.</p>
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