Algebra: domains, systems, inequalities and functions
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| domain restriction/dəˈmeɪn rɪˈstrɪkʃn/ | 定义域限制 | dìng yì yù xiàn zhì |
| simultaneous solution/ˌsɪməlˈteɪnɪəs səˈluːʃn/ | 联立解 | lián lì jiě |
A decision before an answer
- Cancelling x−1 does not make x=1 valid in an expression that originally divided by x−1.
- Your goal: Solve equations while preserving denominator and radical restrictions.
Read the relationship
- 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.
- Analyse inequalities and simultaneous conditions without losing endpoints.
√(x+6)=x has which real solution?
Squaring gives (x−3)(x+2)=0, but only nonnegative 3 works.
Use the defining rule
- 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.
- Evaluate functions and translate verbal relationships into algebra.
−2x<6 is equivalent to:
Divide by negative −2 and reverse the sign.
Check the conditions
- 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.
- Evaluate functions and translate verbal relationships into algebra.
For √(x+2)=x, x≥0. Squaring gives x²−x−2=0, candidates 2 and −1. Only 2 satisfies the original. For (x−1)(x−4)≤0, an upward quadratic is nonpositive on 1≤x≤4, including roots. In the system x+y=10, x−y=2, adding gives 2x=12; x=6 and y=4. For f(t)=2t+1 and g(x)=x², f(g(3))=2·9+1=19, applying g first.
If f(x)=3x−2, f(4)=____.
Substitute into the complete function rule.
Apply the task format
- 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.
- Evaluate functions and translate verbal relationships into algebra.
A transformed equation supplies candidates, not automatic original solutions. Negative inequality multipliers and strict endpoints need explicit checks.
Which answer fits this case?
Solve equations while preserving denominator and radical restrictions
Simplifying (x²−1)/(x−1) removes the original exclusion x=1.
Cancellation preserves the exclusion from the original denominator.
Keep the distinctions
- domain restriction 定义域限制 — A condition identifying inputs on which the original expression is defined.
- simultaneous solution 联立解 — Values satisfying every relation in a system at the same time.
- Solve equations while preserving denominator and radical restrictions.
- Analyse inequalities and simultaneous conditions without losing endpoints.
- Evaluate functions and translate verbal relationships into algebra.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.