Compound Inequalities
Combine two inequalities into one double inequality (AND) or union (OR). Solve by applying the same operation to all three parts simultaneously, then graph the combined solution on a number line.
A temperature must be above 5°C and below 35°C. Can you write this as one inequality using two inequality signs? What do you think the graph looks like?
A compound inequality combines two inequalities. When both must be true (AND), the result is a double inequality: $a < x \leq b$. The solution is the set of values in both ranges.
To solve a double inequality such as $-1 \leq 2x + 3 \leq 9$: apply the same operation to all three parts simultaneously. Treat the middle expression like the "subject" and isolate $x$.
Key facts
- A double inequality $a < x \leq b$ means $x$ must satisfy BOTH inequalities simultaneously.
- Apply operations to all three parts when solving.
- The graph shows a bounded segment between two endpoints.
Concepts
- Why "and" produces an intersection (bounded set) while "or" produces a union (two rays).
- How to use open and closed circles at both endpoints.
- How to translate real-world "safe range" contexts into a double inequality.
Skills
- Solve double inequalities by applying operations to all three parts.
- Graph compound inequalities on a number line with correct circle types.
- Translate a worded range into a compound inequality.
To solve $-1 \leq 2x + 3 \leq 9$: treat it as a single inequality with three parts. Subtract 3 from all three parts, then divide by 2. The variable ends up isolated in the middle.
The answer $-2 \leq x \leq 3$ represents all values from $-2$ to $3$ inclusive. The graph shows a segment with two closed circles.
What to write in your book
- Double inequality: apply each step to all three parts (left, middle, right).
- The result isolates the variable in the middle: $a \leq x \leq b$.
- Draw a segment between the two boundary values using the appropriate circle types.
A compound inequality $-4 < x \leq 2$ graphs as a segment on the number line: open circle at $-4$ (strict, not included), closed circle at $2$ (non-strict, included), shaded between them.
The shaded region represents all values that satisfy both conditions simultaneously.
What to write in your book
- Compound inequality graph = segment between two endpoints (not an arrow).
- Each endpoint gets its own circle type: open ($<$, $>$) or closed ($\leq$, $\geq$).
- The lower boundary must be smaller than the upper boundary for the solution set to exist.
Many real-world situations describe a safe or acceptable range. Translate keywords: "between 6.5 and 8.5" means $6.5 \leq x \leq 8.5$ (if both endpoints are included). Always check whether the boundary values are included.
Example: "Water pH is safe between 6.5 and 8.5 inclusive" → $6.5 \leq \text{pH} \leq 8.5$.
What to write in your book
- "Between a and b inclusive" → $a \leq x \leq b$ (closed circles at both ends).
- "Strictly between a and b" → $a < x < b$ (open circles at both ends).
- Always define your variable before writing the compound inequality.
Worked examples · 3 in a row, reveal as you go
Solve $-1 \leq 2x + 3 \leq 9$.
Graph $-4 < x \leq 2$ on a number line.
Pool water is safe when the pH is between 6.5 and 8.5 (inclusive). Write and graph the compound inequality.
What to write in your book
- Compound AND: solve all three parts together, graph a bounded segment.
- Real-world: define variable, identify lower and upper bounds, check inclusive/exclusive.
- Verify with at least one interior value and both boundary values.
- Solve $0 \leq 3x - 6 < 12$ and graph the solution.
- Write the compound inequality for: closed circle at $-2$, shaded segment, open circle at $4$.
- A school corridor temperature must be between 18°C and 26°C inclusive. Write the inequality.
- Is $x = 4$ in the solution set of $-4 < x \leq 2$? Explain.
The safe temperature range above 5°C AND below 35°C is written as $5 < T < 35$ — an open-circle compound inequality because the boundary values themselves are not "safe".
Earlier you guessed what this might look like. Now explain in full: how does the compound inequality $5 < T < 35$ capture both the minimum and maximum temperature in a single statement?
Pick your answer, then rate your confidence.
Q1. Solve $-3 \leq 2x + 1 \leq 11$ and draw the solution on a number line. (4 marks)
Q2. A fridge must keep food between 1°C and 5°C inclusive. Write the compound inequality, state the boundary values, and explain whether they are included. (3 marks)
Q3. Explain why $5 < x < 2$ has no solution. (2 marks)
📖 Comprehensive answers (click to reveal)
Practice: 1. $2 \leq x < 6$, closed at 2, open at 6. 2. $-2 \leq x < 4$. 3. $18 \leq T \leq 26$. 4. No — $4 > 2$ so $x = 4$ is not in $(-4, 2]$.
Q1 (4 marks): $-3-1 \leq 2x \leq 11-1 \Rightarrow -4 \leq 2x \leq 10$ [1]. Divide by 2: $-2 \leq x \leq 5$ [1]. Closed circle at $-2$, segment, closed circle at $5$ [1]. Check: $x=0$: $-3\leq1\leq11$ ✓ [1].
Q2 (3 marks): $1 \leq T \leq 5$ [1]. Boundary values 1°C and 5°C [1]. Both included (closed circles — "inclusive") [1].
Q3 (2 marks): The left bound $5$ is greater than the right bound $2$ [1], so no value can simultaneously satisfy $x > 5$ and $x < 2$ [1].
Solve compound inequalities and match them to their graphs. Beat the boss to bank a tier.
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