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In the $xy$-plane, line $k$ does not pass through the origin. Which of the following statements, each taken alone, is individually sufficient to conclude that the slope of line $k$ is negative? Select all that apply.
For a non-vertical line with $x$-intercept $p$ and $y$-intercept $q$ (both non-zero since the line avoids the origin), slope $= -\dfrac{q}{p}$.
Statement A โ $p = 2q$: Slope $= -\dfrac{q}{2q} = -\dfrac{1}{2} < 0$. Always negative (as long as $q \neq 0$). Sufficient. โ
Statement B โ $pq > 0$: Then $p$ and $q$ have the same sign. Slope $= -\dfrac{q}{p} = -({\rm same\ sign\ ratio}) < 0$. Always negative. Sufficient. โ
Statement C โ $(a-r)(b-s) < 0$: Slope $= \dfrac{b-s}{a-r}$. Since $(a-r)(b-s) < 0$, numerator and denominator have opposite signs, so slope $< 0$. Sufficient. โ
Statement D โ $y$-intercept $= 0$: This means the line passes through the origin, contradicting the given condition. Invalid / not applicable. โ
Statements A, B, and C are each individually sufficient.
Who is considered the father of 'Liberalism' and advocate for natural rights to life, liberty, and property?
John Locke's 'Two Treatises of Government' laid the groundwork for modern liberal democracy.
Given that $r > s > 0$, compare the two quantities below.
Column A: $\dfrac{rs}{r}$
Column B: $\dfrac{rs}{s}$
Simplify each expression:
- Column A: $\dfrac{rs}{r} = s$
- Column B: $\dfrac{rs}{s} = r$
Since we are given $r > s > 0$, it follows that $r > s$.
Therefore Column B ($r$) is greater than Column A ($s$).
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