openfreebooks-bookgenerator/content/math/linear-inequalities-one-unknown/_index.md

+++
title = "Linear inequalities in one unknown"
description = "Solve, represent, and apply linear inequalities in one unknown using number lines and interval notation."
template = "chapter.html"
weight = 11
[extra]
subject = "math"
chapter_id = "linear-inequalities-one-unknown"
strand = "Number & Algebra"
+++

<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>

<div class="book-prose">
  <h2 id="history" class="book-prose__heading">History</h2>
  <p>
    Ideas behind inequalities are ancient. Greek mathematicians such as <strong>Euclid</strong>
    compared magnitudes and proved strict order relationships long before modern algebraic symbols.
    In India and the Islamic world, later algebra texts worked with bounds and constraints in
    practical problems involving trade, measurement, and inheritance.
  </p>
  <p>
    The modern symbols \(<\) and \(>\) were introduced in the 17th century by
    <strong>Thomas Harriot</strong>. As algebra notation became standard, inequalities became a
    compact way to describe ranges of possible values rather than one exact answer.
    Today, the same language appears in budgeting, engineering safety margins, and optimization.
  </p>

  <h2 id="derivation" class="book-prose__heading">Derivation</h2>

  <h3 id="why-sign-flips" class="book-prose__heading">Why does the sign flip with a negative?</h3>
  <p>
    Suppose \(a < b\). Subtract \(b\) from both sides to get \(a-b < 0\). Multiplying by \(-1\)
    gives \(b-a > 0\), so \(-a > -b\). Geometrically, multiplying by \(-1\) reflects numbers
    across zero on the number line, reversing left-right order. That is why dividing by a negative
    must reverse \(<\) to \(>\) (and \(\leq\) to \(\geq\)).
  </p>

  <h3 id="interval-logic" class="book-prose__heading">Why intervals match inequality endpoints</h3>
  <p>
    A strict inequality such as \(x < c\) excludes \(c\), so interval notation uses an open bracket
    \((\cdot)\). A non-strict inequality such as \(x \leq c\) includes \(c\), so interval notation
    uses a closed bracket \([\cdot]\). This bracket choice encodes exactly the same endpoint logic
    as open and closed dots on number lines.
  </p>

  <h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
  <p>
    Pause and reason before moving on. Discuss with a classmate or write your reasoning in a
    notebook.
  </p>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Solve \(4x - 3 &lt; 9\). Write your answer as both an inequality and an interval.
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Solve \(-5x \geq 20\). Which step forces a sign flip, and what is the final solution?
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Is \(x = -2\) a solution of both \(x \leq -2\) and \(x &lt; -2\)? Explain the difference.
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Solve \(1 \leq 2x + 3 &lt; 9\). Then write the solution in interval notation.
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Write and solve an inequality for this condition: "A package mass \(m\) must be at least
      0.5 kg and less than 2 kg."
    </p>
  </div>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>
      Solve and sketch on a number line: \(x-6 &lt; -10\) or \(2x+1 \geq 7\).
    </p>
  </div>

  <h2 id="applications" class="book-prose__heading">Applications</h2>

  <h3 id="budgeting" class="book-prose__heading">Budgeting and spending limits</h3>
  <p>
    If each notebook costs \$3 and you have at most \$24, the purchase count \(n\) satisfies
    \(3n \leq 24\). Inequalities model spending caps in daily life, event planning, and inventory
    decisions.
  </p>

  <h3 id="safety-limits" class="book-prose__heading">Safety and engineering limits</h3>
  <p>
    Many systems run within allowable ranges, such as pressure \(P \leq P_{\max}\) or temperature
    \(T_{\min} \leq T \leq T_{\max}\). Inequalities represent these safe operating regions directly.
  </p>

  <h3 id="grades-and-thresholds" class="book-prose__heading">Grades and performance thresholds</h3>
  <p>
    A pass condition might be "score at least 50," written as \(s \geq 50\). Scholarship criteria
    and qualification cutoffs are also inequality-based rules.
  </p>

  <h2 id="references" class="book-prose__heading">References</h2>
  <ul>
    <li>Harriot, T. (1631). <em>Artis analyticae praxis</em> (early use of \(<\) and \(>\)).</li>
    <li>Euclid, <em>Elements</em>, Book V (order and comparison of magnitudes).</li>
    <li>Secondary school algebra curricula on linear inequalities and number-line representation.</li>
  </ul>
</div>