Linear Inequalities Pattern - Range

This pattern asks you to express a valid range for a variable using a compound inequality. You'll be given minimum and maximum bounds — from a real-world scenario like temperature limits, acceptable measurements, or observed data — and need to write or select the inequality that captures the full range.

Direct Range from a Minimum and Maximum

The most common Easy version: you're told a quantity has a minimum of $a$ and a maximum of $b$. Write $a \leq x \leq b$.

For a person of a certain age, the target heart rate zone during vigorous exercise is a minimum of 140 beats per minute and a maximum of 175 bpm. Which inequality represents the target heart rate $h$?

A) $h \leq 140$
B) $175 \leq h \leq 315$
C) $h \geq 175$
D) $140 \leq h \leq 175$

Minimum 140 means $h \geq 140$. Maximum 175 means $h \leq 175$. Combine: $140 \leq h \leq 175$. The answer is D. Distractors often show only one bound or add the two numbers together (option B shows 315 = 140 + 175, which is nonsense).

The same pattern appears with phrases like "between X and Y, inclusive," "ranged from X to Y," or "at least X and at most Y." They all mean the same thing.

A litter of newborn puppies was weighed at birth. Their weights ranged from 12 ounces to 16 ounces, inclusive. If $w$ represents the weight of a puppy in ounces, which inequality correctly represents the possible weights?

A) $12 \leq w \leq 16$
B) $w \leq 12$
C) $w \leq 4$
D) $w \leq 16$

"Ranged from 12 to 16, inclusive" → $12 \leq w \leq 16$. The answer is A. Option C uses the difference ($16 - 12 = 4$), which is a common trap — the range of values is not the same as the difference between the bounds. Option D only captures the upper bound.

Rate Times a Range

When a rate is constant and the input (often time) falls within a range, multiply the rate by both bounds to find the range of the output.

A factory machine produces widgets at a constant rate of 80 widgets per hour. The machine operates for more than 7.5 hours but less than 10.5 hours each day. Which inequality represents the total number of widgets, $x$, produced in a day?

A) $80 + 7.5 < x < 80 + 10.5$
B) $80 - 10.5 < x < 80 - 7.5$
C) $\dfrac{80}{10.5} < x < \dfrac{80}{7.5}$
D) $80(7.5) < x < 80(10.5)$

Total widgets = rate $\times$ time. If $7.5 < \text{hours} < 10.5$, then $80(7.5) < x < 80(10.5)$. The answer is D. You multiply (not add, subtract, or divide) the rate by the time bounds. Option A adds instead of multiplying. Option C divides instead of multiplying.

A car has a fuel efficiency of 28 miles per gallon. The driver plans to use more than 9 gallons but fewer than 14 gallons. Which inequality represents the total distance $x$, in miles?

A) $28 \cdot 9 < x < 28 \cdot 14$
B) $28 + 9 < x < 28 + 14$
C) $28 - 14 < x < 28 - 9$
D) $\dfrac{28}{14} < x < \dfrac{28}{9}$

Distance = efficiency $\times$ gallons: $28(9) < x < 28(14)$. The answer is A. Same principle — multiply the rate by both endpoints.

Range with Scaling

At the Medium level, you may need to multiply an hourly range by a number of hours (or a per-minute range by minutes) to find a total.

A manufacturing plant produces at least 150 and at most 165 widgets per hour. Which inequality represents the total number of widgets, $w$, produced during an 8-hour shift?

A) $150 + 8 \leq w \leq 165 + 8$
B) $150 \leq 8w \leq 165$
C) $150 \leq 8 + w \leq 165$
D) $(150)(8) \leq w \leq (165)(8)$

The hourly range is $150 \leq \text{rate} \leq 165$. Over 8 hours, multiply everything by 8: $(150)(8) \leq w \leq (165)(8)$, which is $1200 \leq w \leq 1320$. The answer is D. Option B puts $w$ in the wrong place — it would mean $w$ is the hourly rate, not the total.

A city ordinance states that a residential sprinkler system can use between 1.5 and 2.2 gallons of water per minute, inclusive. Which inequality represents the total amount of water, $g$, if it runs for 45 minutes?

A) $1.5 + 45 \leq g \leq 2.2 + 45$
B) $(1.5)(45) \leq g \leq (2.2)(45)$
C) $1.5 \leq 45g \leq 2.2$
D) $1.5 \leq 45 + g \leq 2.2$

Total = rate $\times$ time. Multiply both bounds by 45: $(1.5)(45) \leq g \leq (2.2)(45)$, i.e., $67.5 \leq g \leq 99$. The answer is B.

Which Range Meets a Criterion?

A twist at Medium difficulty: instead of building the inequality yourself, you're given four possible ranges and must identify which one fully satisfies a stated condition.

A financial analyst considers a stock "low-volatility" if its daily price fluctuation is less than $1.50. Which inequality could represent a low-volatility stock?

A) $1.75 < f < 2.25$
B) $1.00 < f < 2.00$
C) $0.75 < f < 1.25$
D) $1.50 < f < 2.50$

Low-volatility means $f < 1.50$. The entire range must satisfy this — both the lower and upper bounds must be below 1.50.

A) Upper bound 2.25 > 1.50. No. B) Upper bound 2.00 > 1.50. No. C) Upper bound 1.25 < 1.50. Yes — the whole interval is below 1.50. D) Lower bound 1.50, not less than 1.50. No.

The answer is C. The key insight: every value in the range must meet the criterion, so the upper bound of the range must be below the threshold.

A food scientist classifies a bacterial culture as "heat-resistant" if its entire survival range is at or above 80 degrees Celsius. Which inequality could represent a heat-resistant culture?

A) $65 \leq T \leq 75$
B) $78 \leq T \leq 88$
C) $82 \leq T \leq 95$
D) $70 \leq T \leq 79$

Heat-resistant means $T \geq 80$ for the entire range. The lower bound must be at or above 80.

A) Lower bound 65 < 80. No. B) Lower bound 78 < 80. No. C) Lower bound 82 $\geq$ 80. Yes. D) Lower bound 70 < 80. No.

The answer is C. When the criterion is a minimum, check the lower bound of each range. When the criterion is a maximum, check the upper bound.

Triangle Inequality

Hard questions in this pattern often involve the triangle inequality theorem: the sum of any two sides of a triangle must be greater than the third side. Given two sides, the third side must be between their difference and their sum (exclusive).

A sailmaker is cutting a triangular piece of canvas. Two edges are 10 yards and 24 yards long. Which inequality represents the possible lengths, $x$, in yards, for the third edge?

A) $x > 34$
B) $x < 34$
C) $14 < x < 34$
D) $x < 14$ or $x > 34$

By the triangle inequality: the third side must be less than $10 + 24 = 34$ and greater than $24 - 10 = 14$.

So $14 < x < 34$. The answer is C.

The formula: if two sides are $a$ and $b$ (with $a > b$), the third side $x$ satisfies $a - b < x < a + b$. Note the strict inequalities — equality would give a degenerate (flat) triangle.

An engineer is designing a triangular support frame. Two beams have lengths of 5.5 feet and 9.5 feet. Which inequality represents the range of possible lengths, $x$, for the third beam?

A) $4 < x < 15$
B) $x < 15$
C) $x > 15$
D) $x < 4$ or $x > 15$

Lower bound: $9.5 - 5.5 = 4$. Upper bound: $9.5 + 5.5 = 15$. So $4 < x < 15$. The answer is A. Options B and D each miss one of the two bounds.