openfreebooks-bookgenerator/content/math/linear-inequalities-one-unknown/core.html

<div class="book-prose">
  <p>
    A <strong>linear inequality in one unknown</strong> compares a linear expression with a value or
    another linear expression using symbols such as
    \(<\), \(>\), \(\leq\), and \(\geq\). Instead of finding one exact value, we usually find a
    <strong>set of values</strong> that make the statement true.
  </p>

  <h2 id="inequality-symbols" class="book-prose__heading">Inequality symbols</h2>
  <div class="book-table-wrap">
    <table class="book-table">
      <thead>
        <tr>
          <th scope="col">Symbol</th>
          <th scope="col">Meaning</th>
          <th scope="col">Example</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <td>\(<\)</td>
          <td>Strictly less than</td>
          <td>\(x < 4\)</td>
        </tr>
        <tr>
          <td>\(>\)</td>
          <td>Strictly greater than</td>
          <td>\(x > -2\)</td>
        </tr>
        <tr>
          <td>\(\leq\)</td>
          <td>Less than or equal to</td>
          <td>\(x \leq 7\)</td>
        </tr>
        <tr>
          <td>\(\geq\)</td>
          <td>Greater than or equal to</td>
          <td>\(x \geq 0\)</td>
        </tr>
      </tbody>
    </table>
  </div>

  <h2 id="solving" class="book-prose__heading">Solving linear inequalities</h2>
  <p>
    Solve them much like linear equations: expand brackets when needed, collect like terms, then
    isolate the variable.
  </p>
  <p><strong>Example.</strong> Solve \(3x - 5 \leq 16\).</p>
  <p>
    Add \(5\) to both sides: \(3x \leq 21\). Divide by \(3\): \(x \leq 7\).
  </p>

  <h2 id="sign-flip-rule" class="book-prose__heading">The sign-flip rule</h2>
  <p>
    When you multiply or divide <em>both sides</em> of an inequality by a negative number, the
    inequality sign must reverse.
  </p>
  <div class="book-callout book-callout--formula" role="note">
    <p class="book-callout__label">Key rule</p>
    <p class="book-formula">
      \[
      a < b \;\Rightarrow\; -a > -b
      \]
    </p>
  </div>
  <p><strong>Example.</strong> Solve \(-2x + 1 > 9\).</p>
  <p>
    Subtract \(1\): \(-2x > 8\). Divide by \(-2\), so reverse \(>\) to \(<\):
    \(x < -4\).
  </p>

  <h2 id="number-line-intervals" class="book-prose__heading">Number line and interval notation</h2>
  <p>
    The solution set can be written on a number line or in interval notation:
  </p>
  <div class="book-table-wrap">
    <table class="book-table">
      <thead>
        <tr>
          <th scope="col">Inequality</th>
          <th scope="col">Interval notation</th>
          <th scope="col">Endpoint style</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <td>\(x < 3\)</td>
          <td>\((-\infty, 3)\)</td>
          <td>Open at \(3\)</td>
        </tr>
        <tr>
          <td>\(x \leq 3\)</td>
          <td>\((-\infty, 3]\)</td>
          <td>Closed at \(3\)</td>
        </tr>
        <tr>
          <td>\(x > -1\)</td>
          <td>\((-1, \infty)\)</td>
          <td>Open at \(-1\)</td>
        </tr>
        <tr>
          <td>\(x \geq -1\)</td>
          <td>\([-1, \infty)\)</td>
          <td>Closed at \(-1\)</td>
        </tr>
      </tbody>
    </table>
  </div>

  <h2 id="compound-inequalities" class="book-prose__heading">Compound inequalities</h2>
  <p>
    Compound inequalities combine two inequalities with <strong>and</strong> or <strong>or</strong>.
  </p>
  <p><strong>Example (and).</strong> Solve \(2 < x + 1 \leq 6\).</p>
  <p>
    Subtract \(1\) from all three parts:
    \(1 < x \leq 5\). Interval form: \((1, 5]\).
  </p>
  <p><strong>Example (or).</strong> Solve \(x - 4 < -7\) or \(3x + 1 \geq 10\).</p>
  <p>
    First: \(x < -3\). Second: \(3x \geq 9\), so \(x \geq 3\).
    Final set: \((-\infty, -3) \cup [3, \infty)\).
  </p>

  <h2 id="word-problems" class="book-prose__heading">Word problems</h2>
  <p>
    Translate the condition into an inequality, solve, and interpret the answer in context.
  </p>
  <p><strong>Example.</strong> A bus ticket costs \$12. You have at most \$90. How many tickets \(n\)
    can you buy?</p>
  <p>
    \(12n \leq 90 \Rightarrow n \leq 7.5\). Since \(n\) is a whole number of tickets, the maximum is
    \(7\).
  </p>
</div>