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254 lines
11 KiB
HTML
254 lines
11 KiB
HTML
<div class="book-prose">
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<h2 id="history" class="book-prose__heading">History</h2>
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<p>
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For centuries, algebra was written in words. In the 16th and 17th centuries, mathematicians such
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as <strong>François Viète</strong> began to use letters systematically for both unknowns and
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known parameters. That made <strong>formulas</strong> — general relationships such as
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\(v = d/t\) — easy to rearrange without repeating long verbal rules.
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</p>
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<p>
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Rational expressions (fractions built from polynomials) appear whenever you divide one polynomial
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by another. Scientists from <strong>Galileo</strong> onward used ratios and proportionalities in
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motion; later, <strong>Newton’s</strong> laws and electrical laws (e.g. \(V = IR\)) relied on
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changing the subject of a formula to solve for the quantity you can measure.
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</p>
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<p>
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Today, algebraic fractions show up in simplifying models, in calculus limits, and whenever a
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relationship is written as one quantity divided by another.
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</p>
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<h2 id="derivation" class="book-prose__heading">Derivation</h2>
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<h3 id="why-cancel" class="book-prose__heading">Why can we cancel a common factor?</h3>
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<p>
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If \(c \neq 0\), then \(\dfrac{ac}{bc} = \dfrac{a}{b}\) because multiplying numerator and
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denominator by the same non-zero number does not change the value of a numeric fraction; the
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same idea holds when \(c\) is a polynomial factor shared by top and bottom.
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</p>
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<p>
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You <em>cannot</em> cancel in \(\dfrac{a+c}{b+c}\) in general — that would mean, for example,
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\(\dfrac{1+2}{3+2} = \dfrac{1}{3}\), which is false. Only <strong>factors</strong>, not
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<strong>terms</strong>, may be cancelled.
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</p>
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<h3 id="why-reverse-ops" class="book-prose__heading">Why reverse the order of operations?</h3>
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<p>
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A formula like \(v = d/t\) builds \(v\) by first dividing \(d\) by \(t\). To recover \(t\),
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undo division by multiplying: \(vt = d\), then undo multiplication by \(v\) (when \(v \neq 0\))
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to get \(t = d/v\). Each step reverses one operation that was applied to the subject you want.
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</p>
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<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
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<p>
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Pause and reason before continuing. Discuss with a partner or write your work in a notebook.
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</p>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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Simplify \(\dfrac{x^2 - 4}{x + 2}\). What value of \(x\) must be excluded?
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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Can you cancel the \(x\) in \(\dfrac{x + 1}{x}\)? Explain briefly.
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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Write \(\dfrac{3}{x} - \dfrac{1}{2x}\) as a single fraction in simplest form.
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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Make \(d\) the subject of \(v = \dfrac{d}{t}\). State any conditions on \(t\) and \(v\).
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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A wire obeys \(V = IR\). If voltage and resistance are known, which formula gives current?
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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Simplify \(\dfrac{1}{x} + \dfrac{1}{x+1}\). What is the common denominator?
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</p>
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</div>
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<h2 id="applications" class="book-prose__heading">Applications</h2>
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<h3 id="uniform-motion" class="book-prose__heading">Uniform motion</h3>
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<p>
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For constant speed, distance \(d\), speed \(v\), and time \(t\) are related by \(v = d/t\). If a
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train travels \(240\) km in \(3\) h, then \(v = 240/3 = 80\) km/h. If you know \(v\) and \(t\)
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but need distance, rearrange to \(d = vt\).
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</p>
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<div class="math-widget-mount" data-widget="formula-rearrange-demo" data-pagefind-ignore></div>
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<h3 id="density" class="book-prose__heading">Density</h3>
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<p>
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Density \(\rho\) (rho) is mass per unit volume: \(\rho = m/V\). A block of mass \(2.7\) kg with
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volume \(0.001\) m³ has \(\rho = 2.7/0.001 = 2700\) kg/m³. To find mass from \(\rho\) and \(V\),
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use \(m = \rho V\).
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</p>
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<h3 id="pressure-circuits" class="book-prose__heading">Pressure and electric circuits</h3>
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<p>
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Pressure \(P = F/A\) links force \(F\) and area \(A\). Hydraulic and weather models often solve
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for \(F = PA\). In a resistor, \(V = IR\) gives voltage from current and resistance; rearrange
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to \(I = V/R\) when you know voltage and resistance.
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</p>
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<div class="math-widget-mount" data-widget="fraction-simplify-demo" data-pagefind-ignore></div>
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<h2 id="question-bank" class="book-prose__heading">Question bank</h2>
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<p>
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Work each question, then open the solution to check your reasoning. Difficulty increases
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through the sets.
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</p>
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<h3 id="questions-easy" class="book-prose__heading">Easy</h3>
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<details class="book-question">
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<summary class="book-question__prompt">Simplify \(\dfrac{6x}{3x}\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(2\), with \(x \neq 0\).</p>
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<p><strong>Steps:</strong> Cancel the common factor \(3x\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Simplify \(\dfrac{x^2}{x}\) for \(x \neq 0\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(x\).</p>
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<p><strong>Explanation:</strong> Cancel one factor \(x\) from numerator and denominator.</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Make \(d\) the subject of \(v = d/t\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(d = vt\) (\(t \neq 0\) when rearranging from the original).</p>
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<p><strong>Steps:</strong> Multiply both sides by \(t\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Make \(R\) the subject of \(V = IR\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(R = V/I\) (\(I \neq 0\)).</p>
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<p><strong>Steps:</strong> Divide both sides by \(I\).</p>
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</div>
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</details>
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<h3 id="questions-intermediate" class="book-prose__heading">Intermediate</h3>
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<details class="book-question">
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<summary class="book-question__prompt">Simplify \(\dfrac{x^2 - 9}{x + 3}\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(x - 3\), with \(x \neq -3\).</p>
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<p><strong>Steps:</strong> \(x^2 - 9 = (x-3)(x+3)\); cancel \((x+3)\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">\(\dfrac{2}{x} + \dfrac{1}{x} = \) ______</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\dfrac{3}{x}\), \(x \neq 0\).</p>
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<p><strong>Explanation:</strong> Same denominator already.</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Simplify \(\dfrac{x^2 - 1}{x^2 + x}\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\dfrac{x-1}{x}\), \(x \neq 0,\; x \neq -1\).</p>
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<p><strong>Steps:</strong> Factorise; cancel \((x+1)\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Make \(m\) the subject of \(\rho = m/V\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(m = \rho V\) (\(V \neq 0\) in the original).</p>
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<p><strong>Steps:</strong> Multiply both sides by \(V\).</p>
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</div>
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</details>
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<h3 id="questions-difficult" class="book-prose__heading">Difficult</h3>
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<details class="book-question">
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<summary class="book-question__prompt">\(\dfrac{1}{x} + \dfrac{2}{x+1} = \) ______</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\dfrac{3x+1}{x(x+1)}\), \(x \neq 0,\; x \neq -1\).</p>
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<p><strong>Steps:</strong> LCD \(x(x+1)\); numerator \( (x+1) + 2x = 3x+1\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">\(\dfrac{x}{2} \times \dfrac{4}{x^2} = \) ______</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\dfrac{2}{x}\), \(x \neq 0\).</p>
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<p><strong>Steps:</strong> Multiply; cancel a factor \(x\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Simplify \(\dfrac{1}{x-2} - \dfrac{3}{x^2 - 4}\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\dfrac{x-1}{(x-2)(x+2)}\), \(x \neq 2,\; x \neq -2\).</p>
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<p><strong>Steps:</strong> Write \(x^2-4 = (x-2)(x+2)\); common denominator \((x-2)(x+2)\).</p>
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</div>
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</details>
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<h3 id="questions-hardcore" class="book-prose__heading">Hardcore</h3>
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<details class="book-question">
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<summary class="book-question__prompt">\(\dfrac{x^2 - 1}{x+1} \div \dfrac{x-1}{2} = \) ______</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(2\), with \(x \neq -1,\; x \neq 1\).</p>
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<p><strong>Steps:</strong> Factorise \(x^2-1\); multiply by reciprocal; cancel \((x-1)\) and \((x+1)\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Make \(t\) the subject of \(v = \dfrac{d}{t}\) and then find \(t\) when \(d = 150\) km and \(v = 75\) km/h.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(t = d/v = 2\) h.</p>
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<p><strong>Steps:</strong> \(vt = d \Rightarrow t = d/v\); substitute values.</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Simplify fully: \(\dfrac{2x^2 - 8}{x^2 - 4}\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(2\), \(x \neq 2,\; x \neq -2\).</p>
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<p><strong>Steps:</strong> Factorise numerator \(2(x^2-4)\) and denominator \((x-2)(x+2)\); cancel \((x-2)(x+2)\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">A formula is \(P = \dfrac{F}{A}\). The force is \(120\) N and the area is \(0.05\) m². Find \(P\), then make \(F\) the subject and verify your force value.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(P = 120/0.05 = 2400\) Pa; \(F = PA = 2400 \times 0.05 = 120\) N.</p>
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<p><strong>Explanation:</strong> Rearranging checks that the formula is consistent.</p>
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</div>
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</details>
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<h2 id="references" class="book-prose__heading">References</h2>
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<ul>
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<li>Viète, <em>In artem analyticem isagoge</em> (1591) — early systematic use of letters in algebra.</li>
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<li>HKDSE Compulsory Part — Number and Algebra (algebraic fractions, formulas and identities).</li>
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</ul>
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</div>
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