Inequalities, fractions and solution sets
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| intersection/ˌɪntəˈsekʃn/ | giao |
How can we keep the allowed interval without losing a boundary?
- A value must satisfy two conditions at once.
- This lesson studies intersection 交集: The values common to two solution sets, corresponding to AND.
The method
- Adding the same expression preserves an inequality; multiplying or dividing by a negative number reverses it.
- Clear constant denominators using a positive common multiple.
- Factor a quadratic and test the sign on the intervals separated by its roots.
- AND takes the intersection of sets; OR takes their union.
- Distinguish included and excluded endpoints.
Why can (x-1)/(x+2)≥0 not simply be multiplied through by (x+2)?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The reversal rule depends on the multiplier’s sign. The safe method splits at the critical values -2 and 1: the quotient is nonnegative for x<-2 or x≥1, with x=-2 excluded as undefined.
A worked case
For (2x−1)/3−(x+2)/2≥1/6, multiply by 6: 4x−2−3x−6≥1, so x≥9. For (x−2)(x−5)<0, the product is negative between its roots: 2<x<5. Combining with x≥3 gives 3≤x<5, or [3,5). Taking OR with x≤0 gives (−∞,0]∪[3,5). A denominator depending on x needs a separate sign/domain check: (x−1)/(x+2)≥0 has x<−2 or x≥1. The zero numerator at 1 is allowed; x=−2 is undefined.
Inequalities, fractions and solution sets
Adding the same expression preserves an inequality; multiplying or dividing by a negative number reverses it
Match solution-set and boundary decisions to their conditions.
Find the lower bound in (2x−1)/3−(x+2)/2≥1/6.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Multiply by positive 6: 4x−2−3x−6≥1, hence x≥9.
Where it goes wrong
- Do not multiply by an expression of unknown sign as though it were positive.
- For −2x<6, dividing by −2 gives x>−3.
- A root can be included only for a non-strict inequality and only where the expression is defined.
- Infinity is never a real endpoint to include.
Not true: AND means combine every value from both sets. AND means the values in both sets at once, the intersection; OR combines them.
How many integers satisfy 2<x<5 AND x≥3?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The combined interval is [3,5), containing integers 3 and 4.
AND means combine every value from both sets.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
AND means the values in both sets at once, the intersection; OR combines them.
Applying it
- Use a sign table or test value on every interval, then write the whole set.
- The variable-denominator example extends the domain-checking method; B5’s explicit core is linear/quadratic inequalities with brackets and fractions, and combined sets.
- Check x=4 and x=6 in the quadratic case to confirm which side is allowed.
Find the larger boundary root of (x−2)(x−5)=0.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The product is zero at 2 and 5, so the larger root is 5.
Match each set operation to its meaning.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
AND narrows to the overlap while OR pools everything; bracket type records whether each endpoint belongs.
In your course
- AQA A-level Mathematics (7357), specification section B.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Intersection: The values common to two solution sets, corresponding to AND.
- Adding the same expression preserves an inequality; multiplying or dividing by a negative number reverses it.
- Watch for: do not multiply by an expression of unknown sign as though it were positive.