<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Finding Excluded Values in Rational Expressions</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }

    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e293b;
      background: transparent;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: flex-start;
      padding: 8px 12px;
      min-height: 100%;
      user-select: none;
    }

    .container {
      width: 100%;
      max-width: 440px;
      display: flex;
      flex-direction: column;
      align-items: center;
    }

    /* Tab Controls */
    .controls {
      display: flex;
      gap: 6px;
      margin-bottom: 14px;
      width: 100%;
    }

    .btn-step {
      flex: 1;
      border: 1px solid #cbd5e1;
      background: #f8fafc;
      color: #475569;
      padding: 8px 4px;
      font-size: 13px;
      font-weight: 600;
      border-radius: 6px;
      cursor: pointer;
      text-align: center;
      transition: background 0.2s ease, border-color 0.2s ease, color 0.2s ease;
    }

    .btn-step:hover {
      background: #f1f5f9;
      color: #1e293b;
    }

    .btn-step.active {
      background: #2563eb;
      border-color: #2563eb;
      color: #ffffff;
    }

    /* Stage without card borders */
    .stage {
      position: relative;
      width: 100%;
      min-height: 250px;
      display: flex;
      align-items: center;
      justify-content: center;
    }

    .panel {
      position: absolute;
      inset: 0;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      opacity: 0;
      pointer-events: none;
      transition: opacity 0.3s ease;
      padding: 4px 0;
    }

    .panel.active {
      opacity: 1;
      pointer-events: auto;
      position: relative;
    }

    /* Mathematical Typography */
    .math {
      font-family: "Cambria Math", "STIX Two Math", "Times New Roman", Georgia, serif;
      font-size: 21px;
      line-height: 1.2;
      color: #0f172a;
    }

    var {
      font-style: italic;
      font-family: inherit;
    }

    .fraction {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      padding: 0 4px;
    }

    .numerator {
      border-bottom: 2px solid #334155;
      padding: 2px 4px;
    }

    .denominator {
      padding: 3px 4px 1px 4px;
    }

    .math-pill {
      background: #f1f5f9;
      border-radius: 8px;
      padding: 8px 16px;
      display: inline-flex;
      align-items: center;
      justify-content: center;
    }

    .focus-pill {
      background: #fef2f2;
      border: 1.5px solid #f87171;
      border-radius: 8px;
      padding: 8px 16px;
    }

    .arrow-indicator {
      color: #64748b;
      font-size: 16px;
      font-weight: bold;
      margin: 4px 0;
    }

    /* Step 2 Split Layout */
    .split-row {
      display: flex;
      gap: 12px;
      width: 100%;
      justify-content: center;
      margin: 8px 0;
    }

    .sub-box {
      flex: 1;
      max-width: 180px;
      background: #fff5f5;
      border: 1.5px dashed #f87171;
      border-radius: 8px;
      padding: 10px 8px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 6px;
    }

    .res-val {
      font-weight: 700;
      color: #dc2626;
      font-size: 20px;
    }

    /* Badges and tags */
    .badge {
      font-size: 12px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      padding: 4px 10px;
      border-radius: 12px;
    }

    .badge-red {
      color: #b91c1c;
      background: #fee2e2;
    }

    .badge-blue {
      color: #1d4ed8;
      background: #dbeafe;
    }

    /* Step 3 Number Line */
    .number-line-wrap {
      width: 100%;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 8px;
    }

    .number-line-svg {
      width: 100%;
      height: 100px;
      overflow: visible;
    }

    .domain-desc {
      font-size: 14px;
      font-weight: 600;
      color: #1e293b;
      text-align: center;
    }

    @media (max-width: 380px) {
      .math { font-size: 18px; }
      .res-val { font-size: 18px; }
      .btn-step { font-size: 12px; padding: 6px 2px; }
    }
  </style>
</head>
<body>

  <div class="container">
    <div class="controls">
      <button class="btn-step active" onclick="setStep(0)">1. Factor</button>
      <button class="btn-step" onclick="setStep(1)">2. Solve = 0</button>
      <button class="btn-step" onclick="setStep(2)">3. Domain</button>
    </div>

    <div class="stage">
      
      <div class="panel active" id="panel-0">
        <div class="math-pill math">
          <div class="fraction">
            <span class="numerator"><var>x</var> + 2</span>
            <span class="denominator"><var>x</var><sup>2</sup> − 2<var>x</var> − 15</span>
          </div>
        </div>

        <div class="arrow-indicator">↓ Factor Denominator</div>

        <div class="math-pill math">
          <div class="fraction">
            <span class="numerator"><var>x</var> + 2</span>
            <span class="denominator">(<var>x</var> − 5)(<var>x</var> + 3)</span>
          </div>
        </div>

        <span class="badge badge-red">Denominator Cannot Equal 0</span>
      </div>

      
      <div class="panel" id="panel-1">
        <div class="focus-pill math">
          (<var>x</var> − 5)(<var>x</var> + 3) = 0
        </div>

        <div class="split-row">
          <div class="sub-box">
            <div class="math"><var>x</var> − 5 = 0</div>
            <div class="arrow-indicator" style="font-size: 12px; margin: 0;">↓</div>
            <div class="math res-val"><var>x</var> = 5</div>
          </div>
          <div class="sub-box">
            <div class="math"><var>x</var> + 3 = 0</div>
            <div class="arrow-indicator" style="font-size: 12px; margin: 0;">↓</div>
            <div class="math res-val"><var>x</var> = −3</div>
          </div>
        </div>

        <span class="badge badge-red">Excluded Values: x = 5, x = −3</span>
      </div>

      
      <div class="panel" id="panel-2">
        <div class="domain-desc">
          Domain: All Real Numbers Except <span style="color:#dc2626; font-weight:700;">−3</span> and <span style="color:#dc2626; font-weight:700;">5</span>
        </div>

        <div class="number-line-wrap">
          <svg class="number-line-svg" viewBox="0 0 360 80">
            <defs>
              
              <marker id="arrow-start" viewBox="0 0 10 10" refX="2" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
                <path d="M 10 0 L 0 5 L 10 10 z" fill="#2563eb"/>
              </marker>
              <marker id="arrow-end" viewBox="0 0 10 10" refX="8" refY="5" markerWidth="6" markerHeight="6" orient="auto">
                <path d="M 0 0 L 10 5 L 0 10 z" fill="#2563eb"/>
              </marker>
            </defs>

            
            <line x1="12" y1="42" x2="348" y2="42" stroke="#cbd5e1" stroke-width="2" />

            
            
            
            <line x1="12" y1="42" x2="95" y2="42" stroke="#2563eb" stroke-width="5" marker-start="url(#arrow-start)" />
            
            <line x1="95" y1="42" x2="295" y2="42" stroke="#2563eb" stroke-width="5" />
            
            <line x1="295" y1="42" x2="348" y2="42" stroke="#2563eb" stroke-width="5" marker-end="url(#arrow-end)" />

            
            <g stroke="#64748b" stroke-width="1.5">
              <line x1="20" y1="37" x2="20" y2="47" />
              <line x1="70" y1="38" x2="70" y2="46" />
              <line x1="120" y1="38" x2="120" y2="46" />
              <line x1="170" y1="35" x2="170" y2="49" stroke="#0f172a" stroke-width="2" />
              <line x1="220" y1="38" x2="220" y2="46" />
              <line x1="270" y1="38" x2="270" y2="46" />
              <line x1="320" y1="37" x2="320" y2="47" />
            </g>

            
            <g font-size="11" fill="#64748b" text-anchor="middle" font-family="system-ui, sans-serif" font-weight="500">
              <text x="20" y="62">-6</text>
              <text x="70" y="62">-4</text>
              <text x="120" y="62">-2</text>
              <text x="170" y="62" font-weight="700" fill="#0f172a">0</text>
              <text x="220" y="62">2</text>
              <text x="270" y="62">4</text>
              <text x="320" y="62">6</text>
            </g>

            
            <circle cx="95" cy="42" r="6" fill="#ffffff" stroke="#dc2626" stroke-width="3" />
            <text x="95" y="24" font-size="12" font-weight="700" fill="#dc2626" text-anchor="middle" font-family="system-ui, sans-serif">x ≠ -3</text>

            
            <circle cx="295" cy="42" r="6" fill="#ffffff" stroke="#dc2626" stroke-width="3" />
            <text x="295" y="24" font-size="12" font-weight="700" fill="#dc2626" text-anchor="middle" font-family="system-ui, sans-serif">x ≠ 5</text>
          </svg>
        </div>

        <span class="badge badge-blue">Open Red Circles = Excluded Points</span>
      </div>
    </div>
  </div>

  <script>
    let currentStep = 0;
    const buttons = document.querySelectorAll('.btn-step');
    const panels = [
      document.getElementById('panel-0'),
      document.getElementById('panel-1'),
      document.getElementById('panel-2')
    ];

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      currentStep = index;
      buttons.forEach((btn, i) => btn.classList.toggle('active', i === index));
      panels.forEach((p, i) => p.classList.toggle('active', i === index));
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    // Auto-advance every 4.5 seconds
    let timer = setInterval(() => {
      setStep((currentStep + 1) % 3);
    }, 4500);

    // Pause cycling on user interaction
    buttons.forEach(btn => {
      btn.addEventListener('click', () => {
        clearInterval(timer);
      });
    });
  </script>
</body>
</html>

Identifying Excluded Values and Domain Restrictions in Rational Expressions

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Excluded Values in Simplified Expressions</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: transparent;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }

    .diagram-flow {
      width: 100%;
      max-width: 620px;
      display: flex;
      flex-direction: column;
      gap: 12px;
    }

    .boxes-row {
      display: flex;
      flex-direction: row;
      gap: 12px;
      width: 100%;
    }

    @media (max-width: 480px) {
      .boxes-row {
        flex-direction: column;
      }
    }

    .math-card {
      flex: 1;
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 12px;
      padding: 14px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 10px;
      box-shadow: 0 2px 6px rgba(0, 0, 0, 0.04);
      position: relative;
    }

    .card-title {
      font-size: 13px;
      font-weight: 700;
      letter-spacing: 0.5px;
      text-transform: uppercase;
      color: #64748b;
    }

    .math-display {
      font-family: "Cambria Math", "KaTeX_Math", "Times New Roman", serif;
      font-size: 18px;
      display: flex;
      align-items: center;
      justify-content: center;
      min-height: 52px;
    }

    .fraction {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      padding: 0 4px;
    }

    .numerator {
      border-bottom: 2px solid #1e2945;
      padding-bottom: 3px;
      margin-bottom: 2px;
    }

    .denominator {
      padding-top: 2px;
    }

    /* Strike-through styling */
    .strike {
      position: relative;
      display: inline-block;
      color: #94a3b8;
    }

    .strike::after {
      content: '';
      position: absolute;
      left: -2px;
      top: 50%;
      width: calc(100% + 4px);
      height: 2.5px;
      background-color: #ef4444;
      transform: rotate(-12deg);
      border-radius: 2px;
    }

    .highlight-term {
      color: #2563eb;
      font-weight: bold;
    }

    .equals {
      margin: 0 6px;
      color: #64748b;
    }

    .badge {
      display: inline-flex;
      align-items: center;
      gap: 6px;
      padding: 5px 10px;
      border-radius: 6px;
      font-size: 13px;
      font-weight: 700;
    }

    .badge-error {
      background: #fef2f2;
      border: 1.5px solid #f87171;
      color: #dc2626;
    }

    .badge-check {
      background: #f0fdf4;
      border: 1.5px solid #86efac;
      color: #16a34a;
    }

    .warning-banner {
      width: 100%;
      background: #fffbeb;
      border: 1.5px solid #f59e0b;
      border-radius: 10px;
      padding: 10px 14px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      font-size: 13.5px;
      font-weight: 700;
      color: #b45309;
      text-align: center;
      box-shadow: 0 2px 5px rgba(245, 158, 11, 0.1);
    }

    .action-btn {
      background: #f1f5f9;
      border: 1.5px solid #cbd5e1;
      color: #334155;
      padding: 6px 14px;
      border-radius: 6px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 6px;
      transition: background 0.2s, border-color 0.2s;
    }

    .action-btn:hover {
      background: #e2e8f0;
      border-color: #94a3b8;
    }

    .test-result {
      width: 100%;
      font-size: 12.5px;
      font-family: "Cambria Math", monospace, serif;
      background: #f8fafc;
      border-radius: 6px;
      padding: 6px 10px;
      text-align: center;
      border: 1px dashed #cbd5e1;
      display: none;
    }

    .test-result.visible {
      display: block;
    }
  </style>
</head>
<body>

  <div class="diagram-flow">
    <div class="boxes-row">
      
      <div class="math-card">
        <div class="card-title">1. Original Expression</div>
        <div class="math-display">
          <span class="fraction">
            <span class="numerator">(x − 4)(x + 1)</span>
            <span class="denominator highlight-term">x − 4</span>
          </span>
        </div>
        <div class="badge badge-error">
          <span>⚠️ Excluded: x = 4</span>
        </div>
        <div id="test1" class="test-result">
          If x = 4: <span style="color:#dc2626;">(0)(5) / 0 = Undefined!</span>
        </div>
      </div>

      
      <div class="math-card">
        <div class="card-title">2. Simplified</div>
        <div class="math-display">
          <span class="fraction">
            <span class="numerator"><span class="strike">(x − 4)</span>(x + 1)</span>
            <span class="denominator"><span class="strike">x − 4</span></span>
          </span>
          <span class="equals">=</span>
          <span class="highlight-term">x + 1</span>
        </div>
        <div class="badge badge-error">
          <span>⚠️ Still Excluded: x ≠ 4</span>
        </div>
        <div id="test2" class="test-result">
          Domain: <span style="color:#dc2626;">All real x except x = 4</span>
        </div>
      </div>
    </div>

    
    <div class="warning-banner">
      <span>⚡ Warning: x = 4 is STILL excluded even though the term canceled out!</span>
    </div>

    
    <div style="display:flex; justify-content:center;">
      <button id="toggleBtn" class="action-btn" onclick="toggleTest()">
        <span>🔍 Test plugging in x = 4</span>
      </button>
    </div>
  </div>

  <script>
    let tested = false;
    function toggleTest() {
      tested = !tested;
      const res1 = document.getElementById('test1');
      const res2 = document.getElementById('test2');
      const btn = document.getElementById('toggleBtn');

      if (tested) {
        res1.classList.add('visible');
        res2.classList.add('visible');
        btn.innerHTML = '<span>✕ Hide Test</span>';
      } else {
        res1.classList.remove('visible');
        res2.classList.remove('visible');
        btn.innerHTML = '<span>🔍 Test plugging in x = 4</span>';
      }
    }
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="1:1">
  <title>Rational Expression Domain Rule</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }

    body {
      background-color: #f8f9fa;
      color: #1e2945;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 16px;
      user-select: none;
    }

    .diagram-area {
      width: 100%;
      max-width: 340px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
    }

    /* Fraction block */
    .fraction-container {
      display: flex;
      flex-direction: column;
      align-items: center;
      width: 100%;
      position: relative;
    }

    .math-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      padding: 8px 12px;
      border-radius: 12px;
      transition: background-color 0.3s ease, transform 0.2s ease;
    }

    .numerator-row {
      background: #f0fdf4;
      border: 1.5px dashed #86efac;
    }

    .denominator-row {
      background: #fef2f2;
      border: 1.5px dashed #fca5a5;
      position: relative;
    }

    .math-symbol {
      font-family: "Cambria Math", "KaTeX_Math", "Times New Roman", serif;
      font-size: 26px;
      font-weight: 700;
      font-style: italic;
      color: #1e2945;
      min-width: 60px;
    }

    .fraction-divider {
      width: 100%;
      height: 4px;
      background: #1e2945;
      border-radius: 2px;
      margin: 6px 0;
    }

    /* Badges */
    .badge {
      display: inline-flex;
      align-items: center;
      gap: 6px;
      font-size: 13px;
      font-weight: 600;
      padding: 6px 10px;
      border-radius: 999px;
      line-height: 1;
    }

    .badge-green {
      background: #dcfce7;
      color: #15803d;
      box-shadow: 0 1px 3px rgba(22, 101, 52, 0.1);
    }

    .badge-red {
      background: #fee2e2;
      color: #b91c1c;
      box-shadow: 0 1px 3px rgba(185, 28, 28, 0.1);
    }

    .badge-icon {
      font-size: 14px;
      font-style: normal;
    }

    /* Neon Warning Callout */
    .callout {
      position: relative;
      background: #fff;
      border: 2px solid #ef4444;
      border-radius: 10px;
      padding: 10px 14px;
      color: #991b1b;
      font-size: 14px;
      font-weight: 700;
      text-align: center;
      box-shadow: 0 0 12px rgba(239, 68, 68, 0.25), 0 2px 6px rgba(0, 0, 0, 0.05);
      animation: pulse-glow 2.5s infinite ease-in-out;
      width: 100%;
    }

    .callout::before {
      content: "";
      position: absolute;
      top: -8px;
      left: 36px;
      width: 12px;
      height: 12px;
      background: #fff;
      border-top: 2px solid #ef4444;
      border-left: 2px solid #ef4444;
      transform: rotate(45deg);
    }

    @keyframes pulse-glow {
      0%, 100% {
        box-shadow: 0 0 8px rgba(239, 68, 68, 0.2);
      }
      50% {
        box-shadow: 0 0 16px rgba(239, 68, 68, 0.45);
      }
    }

    /* Interactive Demonstration */
    .interactive-card {
      width: 100%;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px 14px;
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 8px;
    }

    .toggle-btn {
      background: #1e2945;
      color: #ffffff;
      border: none;
      padding: 6px 12px;
      font-size: 13px;
      font-weight: 600;
      border-radius: 6px;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 6px;
      transition: background-color 0.2s ease, opacity 0.2s ease;
    }

    .toggle-btn:active {
      opacity: 0.85;
    }

    .result-text {
      font-family: "Cambria Math", "KaTeX_Math", "Times New Roman", serif;
      font-size: 15px;
      font-weight: 700;
      color: #1e2945;
    }

    .result-text.error {
      color: #dc2626;
    }
  </style>
</head>
<body>

  <div class="diagram-area">
    
    <div class="fraction-container">
      
      
      <div class="math-row numerator-row">
        <span class="math-symbol">P(x)</span>
        <span class="badge badge-green">
          <span class="badge-icon">✓</span> Allowed to be zero
        </span>
      </div>

      <div class="fraction-divider"></div>

      
      <div class="math-row denominator-row">
        <span class="math-symbol">Q(x)</span>
        <span class="badge badge-red">
          <span class="badge-icon">⚠</span> Q(x) ≠ 0
        </span>
      </div>

    </div>

    
    <div class="callout">
      ⚡ Zero here breaks the expression!
    </div>

    
    <div class="interactive-card">
      <button class="toggle-btn" id="testBtn">
        <span>Try Q(x) = 0</span>
      </button>
      <div class="result-text" id="resultLabel">12 ÷ 3 = 4</div>
    </div>
  </div>

  <script>
    const btn = document.getElementById('testBtn');
    const result = document.getElementById('resultLabel');
    let isZero = false;

    btn.addEventListener('click', () => {
      isZero = !isZero;
      if (isZero) {
        btn.innerHTML = '<span>Reset</span>';
        btn.style.background = '#dc2626';
        result.innerHTML = '12 ÷ 0 = <span style="font-family:system-ui;font-weight:800;">UNDEFINED 💥</span>';
        result.classList.add('error');
      } else {
        btn.innerHTML = '<span>Try Q(x) = 0</span>';
        btn.style.background = '#1e2945';
        result.textContent = '12 ÷ 3 = 4';
        result.classList.remove('error');
      }
    });
  </script>

</body>
</html>

OGM.9.ALG.2

Aligned learning resources

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