Compound Inequalities

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.

Double inequality a ≤ bx + c ≤ d: apply the same operation to all three parts simultaneously. Subtract c from all three, then divide all three by b. If b is negative, flip both inequality signs. Presents the answer as a range.

Pause, copy the three-part operation rule for double inequalities: apply each algebraic step to all three parts simultaneously (e.g. −1 ≤ 2x + 3 ≤ 9: subtract 3 from all three → −4 ≤ 2x ≤ 6; divide all by 2 → −2 ≤ x ≤ 3) into your book.

We just saw that double inequalities like −1 ≤ 2x + 3 ≤ 9 are solved by applying the same operation to all three parts simultaneously, subtract 3 from all three, then divide all by 2. That raises a question: after solving to get −2 ≤ x ≤ 3, how do you show this as a graph on a number line? This card answers it → draw two boundary circles at −2 and 3, shade the segment between them, and choose open or closed circles based on whether each boundary is included.

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.

Compound inequality graph: two boundary values, each with its own open or closed circle. The segment between them is shaded. Open circle = strict; closed = non-strict. Read by identifying both endpoints and their inclusion.

Pause, copy the compound inequality graph rules: two boundary values, each with its own open (strict) or closed (non-strict) circle, with the segment between them shaded, and the step of reading back both endpoints and their inclusion status into your book.

We just saw graphing compound inequalities as a shaded segment between two boundary circles on the number line. That raises a question: compound inequalities often arise from real-world descriptions like "the temperature must be between 18°C and 24°C", how do you convert that language into a correct inequality? This card answers it → key phrase translations: "between a and b" → a ≤ x ≤ b (check whether the boundary is included); "at least a and at most b" → a ≤ x ≤ b; note that "between" alone is ambiguous about whether endpoints are included.

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" → $6.5 \leq \text{pH} \leq 8.5$.

Real-world ranges: ‘between a and b’ → a ≤ x ≤ b; ‘between a and b exclusive’ → a < x < b; ‘at least a and at most b’ → a ≤ x ≤ b. Read the boundary inclusion carefully, ‘between’ alone is ambiguous in everyday English.

Pause, copy the phrase-to-inequality translations ("between a and b" → a ≤ x ≤ b; "at least a and at most b" → a ≤ x ≤ b; "more than a and less than b" → a < x < b) and the ambiguity warning that "between" alone does not specify whether endpoints are included into your book.

1. Solve $0 \leq 3x - 6 < 12$ and graph the solution.
2. Write the compound inequality for: closed circle at $-2$, shaded segment, open circle at $4$.
3. A school corridor temperature must be between 18°C and 26°C inclusive. Write the inequality.
4. Is $x = 4$ in the solution set of $-4 < x \leq 2$? Explain.