Q.L-algebra · Algebra: domains, systems, inequalities and functions

الممارسة الصفية الأصلية ضمن أنواع المهام الاختبارية المختصرة المراجعة. لا يحاكي هذا الامتحان التكيفي ولا يتوقع درجة مقيسة أو كتابة. تظل النماذج الرسمية العامة السابقة كتمارين مكشوفة؛ لا تزال إصداراتها ومصدر الكتابة سارية المفعول.

• حل المعادلات مع الحفاظ على قيود المقام والجذر
• تحليل المتباينات والشروط المتزامنة دون فقدان النقاط الطرفية
• Evaluate functions and translate verbal relationships into algebra

domain restriction: A condition identifying inputs on which the original expression is defined.

simultaneous solution: Values satisfying every relation in a system at the same time.

Write restrictions before manipulating an equation. A denominator must be nonzero; a real even root needs a nonnegative argument. Cancelling factors preserves the expression only on its original domain. Squaring can introduce candidates because a square loses sign information; verify every candidate in the unsquared relation.

Solving a system requires the same ordered pair to satisfy all relations. Substitution or elimination can expose one, no or infinitely many linear solutions. For an inequality, multiplication or division by a negative reverses direction. For a factored quadratic inequality, order roots and test interval signs, including endpoints only when equality is allowed.

A function assigns one output to each permitted input. Evaluate the inner function first in a composition, preserving domain restrictions. Read a slope as change in output per input unit and an intercept as the output at input zero only when that input is meaningful. A verbal model should name variables and keep quantities in compatible units.

For Quantitative Comparison, preserve all allowed values. A variable with x²=4 could be −2 or 2 unless constrained. One counterexample can reject a universal relationship; several favourable substitutions cannot prove it for every real value. Algebraic structure can establish a relationship without exhaustive numerical sampling.

بالنسبة لـ √(x+2)=x، x≥0. التربيع يعطي x²−x−2=0، المرشحات 2 و −1. فقط 2 يحقق الأصلي. بالنسبة لـ (x−1)(x−4)≤0، التربيعي المتجه للأعلى يكون غير موجب على 1≤x≤4، بما في ذلك الجذور. في النظام x+y=10، x−y=2، الجمع يعطي 2x=12؛ x=6 و y=4. بالنسبة لـ f(t)=2t+1 و g(x)=x²، f(g(3))=2·9+1=19، بتطبيق g أولاً.

المعادلة المحولة توفر مرشحات، وليس حلولاً أصلية تلقائية. ضروب المتباينات السالبة والنقاط الطرفية الصارمة تحتاج فحوصات صريحة.

Solve $(x^2-4)/(x-2)=5$ on its original real domain. Solve $(x+1)(x-3)\le0$ and find $f(g(2))$ for $f(t)=t^2$, $g(x)=x-3$.

النموذج والتبرير

The rational equation excludes x=2. For allowed x it reduces to x+2=5, giving x=3, which substitutes correctly. The quadratic is nonpositive between its roots, including them: $-1\le x\le3$. The composition applies g first: $g(2)=-1$ and $f(g(2))=1$. Cancelling a factor does not restore excluded inputs.

For real a, classify the system $x+y=4$, $2x+ay=8$ as having one, none or infinitely many solutions. Compare with the second equation $2x+ay=9$. Give solutions in every consistent case.

تحقق بعد المحاولة

Subtract twice the first equation. In the first system $(a-2)y=0$: for a not equal to two, y=0,x=4, uniquely. For a=2, both equations are the same relation, giving every pair $(4-t,t)$. There is no inconsistent case in this first system. With right side nine, $(a-2)y=1$. For a not equal to two, $y=1/(a-2)$ and $x=4-1/(a-2)$, uniquely. For a=2 it demands 0=1, so no solution. A zero coefficient row means inconsistency only when the right side is nonzero.