Inequalities
Introduced| English | Meaning |
|---|---|
| inequality/ɪniːˈkwɒlɪti/ | A mathematical statement comparing expressions using relations such as less than or greater than. |
| intervals/ˈɪntəvlz/ | A continuous set of real numbers between specified bounds, which may include or exclude its endpoints. |
| number line/ˈnʌmbə laɪn/ | A line showing numbers in order, used for inequalities and directed numbers. |
| region/ˈriːdʒn/ | A specified part of a space, surface or area. |
Speed limits and number ranges
- A road sign says "Speed ≤ 30 mph". That's an inequality 不等式 — a range of allowed values, not a single answer.
- Inequalities describe intervals 区间: a set of values, not one value. Solving them is like solving equations, with one crucial twist.
Solving inequalities (Extended) (Extended)
- Solve exactly like an equation — but reverse the sign ($<$ becomes $>$, and vice versa) if you multiply or divide by a negative number.
Solve $3x + 1 < 10$: subtract 1 → $3x < 9$ → divide by 3 → $x < 3$.
The sign flip. Solve $-2x > 6$: divide by $-2$ and flip → $x < -3$ (not $x > -3$). Forgetting to flip is the most common mistake.
$-\tfrac13 \leqslant x < 3$: a closed circle includes the end value, an open circle excludes it
Inequalities
y = ax + b
An inequality asks where the line is above or below a value.
You must reverse the inequality sign when you:
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Multiplying or dividing by a negative flips < to > (and vice versa).
Solve 3x + 1 < 10. What is the upper bound for x?
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3x < 9 → x < 3. The upper bound is 3 (not included).
Solving −2x > 6 gives x > −3.
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Dividing by −2 and flipping gives x < −3, not x > −3.
Solving double inequalities (Extended)
- $-3 \leq 3x - 2 < 7$: apply the same operation to all three parts.
- Add 2: $-1 \leq 3x < 9$. Divide by 3: $-\dfrac{1}{3} \leq x < 3$.
Solve −3 ≤ 3x − 2 < 7. The solution is −1/3 ≤ x < ? What is the upper bound?
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Add 2: −1 ≤ 3x < 9; divide by 3: −1/3 ≤ x < 3.
On a number line 数轴
- Open circle (○) for $<$ or $>$ — the end value is not included.
- Closed circle (●) for $\leq$ or $\geq$ — the end value is included.
Closed circle at $-\frac{1}{3}$ (included), open circle at $3$ (excluded). The shaded region 区域 is the solution.
On a number line, the symbol ≤ is drawn with a:
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≤ and ≥ include the end value, shown by a closed circle; < and > use an open circle.
Match each inequality symbol to its number-line drawing.
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< and > use open circles (end value excluded); ≤ and ≥ use closed circles (included).
Inequality regions (Extended)
- An inequality in two variables ($y > 2x + 1$) shades a region on the coordinate plane.
- Broken (dashed) boundary line for strict inequalities ($<$ or $>$).
- Solid boundary line for $\leq$ or $\geq$.
Core: interpret before solving
- For $-3\le x<2$, mark a filled circle at $-3$, an open circle at 2 and shade between. The integer values are $-3,-2,-1,0,1$.
- A parcel rule $2\le m<5$ kg includes 2 kg but excludes 5 kg. Core questions ask you to interpret and represent such ranges; solving algebraic inequalities belongs to Extended.
How many integers satisfy −3 ≤ x < 2?
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The integers are −3, −2, −1, 0 and 1.
You've got it
- solve like an equation, but flip the sign when multiplying/dividing by a negative
- $-3 \leq 3x - 2 < 7 \Rightarrow -\dfrac{1}{3} \leq x < 3$
- number line: open circle excludes, closed circle includes the end
- 2-variable inequalities shade a region (dashed boundary for strict, solid otherwise)