Before you start, make sure you know: Rational numbers (Math 8 (Pre-Algebra))
A roller coaster sign says: "Riders must be between 48 and 74 inches tall, inclusive." A pool needs a safe pH range. A machinist has a tolerance on a part. All of these are pieces of the number line. In this section you'll learn a short, exact notation for them: inequalities, graphs, and interval notation.
Intervals
An interval is the set of all real numbers between two endpoints. Two things matter: where the piece starts and ends, and whether each endpoint is included.
On a number line, a closed circle (filled) means "included" and an open circle means "not included." This matches the inequality symbols: $\le$ and $\ge$ include the endpoint, $<$ and $>$ do not.
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Observe the graphical representation of the following set of real numbers and express it using interval notation and as an inequality.
Observe the graphical representation of the following set of real numbers and express it using interval notation and as an inequality.
Some sets go on forever in one direction: "at least 48 inches" or "below freezing." These are rays. One endpoint is a number. The other is $\infty$ or $-\infty$.
Translating English into inequalities and intervals is mostly about a few key phrases:
Words
Inequality
Interval
more than $a$
$x > a$
$(a, \infty)$
at least $a$
$x \ge a$
$[a, \infty)$
less than $b$
$x < b$
$(-\infty, b)$
at most $b$ (or no more than $b$)
$x \le b$
$(-\infty, b]$
between $a$ and $b$
$a < x < b$
$(a, b)$
from $a$ to $b$, inclusive
$a \le x \le b$
$[a, b]$
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Write as a ray the set of all real numbers that are less than or equal to $7$.
Write as a ray the set of all real numbers that are less than or equal to $7$.
Match each of the following phrases with its corresponding mathematical interval: "Temperatures below zero." "Ages between 12 and 16 years, both inclusive." "Prices…
Match each of the following phrases with its corresponding mathematical interval:
The coaster sign gives the heights $[48, 74]$. We can describe the same interval another way: by its middle and how far it reaches to each side. Some problems call this a neighborhood, written $E(a, r)$.
To switch from an interval $(p, q)$ to center-and-radius form:
center = midpoint $= \dfrac{p + q}{2}$
radius = distance from the center to an endpoint $= \dfrac{q - p}{2}$ (half the length)
Why? The center is the point equally far from both ends, which is the midpoint. The radius is that equal distance.
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Write the open interval $(2, 10)$ as a neighborhood $E(a, r)$.
Write the open interval $(2, 10)$ as a neighborhood $E(a, r)$.
In the field of nutrition, the Body Mass Index (BMI) is considered healthy if it is in the interval $[18.5, 24.9]$. Express this range as a closed neighborhood,…
In the field of nutrition, the Body Mass Index (BMI) is considered healthy if it is in the interval $[18.5, 24.9]$. Express this range as a closed neighborhood, calculating its center and radius first.
The absolute value $|x - a|$ is the distance between $x$ and $a$ on the number line. (Some problems say modulus, which is the same thing as absolute value.) Distance is exactly what a center and radius describe, so absolute value inequalities and intervals are two ways to say the same thing.
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Express as an inequality and as an interval the set of real numbers whose distance from zero is strictly less than $5$.
Express as an inequality and as an interval the set of real numbers whose distance from zero is strictly less than $5$.
Determine the value of the parameter $k$ so that the solution interval of the inequality $|x - k| \le 5$ has the number $12$ as its upper bound. Once $k$ is calculated,…
Determine the value of the parameter $k$ so that the solution interval of the inequality $|x - k| \le 5$ has the number $12$ as its upper bound. Once $k$ is calculated, state what the lower bound of the interval is and express it as a neighborhood.
In real life, "target $\pm$ error" is everywhere: parts in a factory, thermometers, measuring cups. The phrase "the value is $a$ with error at most $r$" is exactly $|x - a| \le r$.
Now the reverse situation: an instrument that is itself imprecise. If a thermometer reads $m$ and has a maximum error of $e$, the true value is somewhere in $[m - e,\ m + e]$. To be certain the true value is in a safe interval, that whole interval has to fit inside the safe one.
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A machine produces steel shafts whose ideal length is $120\text{ mm}$. Quality control allows a maximum error of $0.5\text{ mm}$. Express the range of permitted lengths…
A machine produces steel shafts whose ideal length is $120\text{ mm}$. Quality control allows a maximum error of $0.5\text{ mm}$. Express the range of permitted lengths as a neighborhood, as an inequality with a modulus, and finally as an interval.
The share price of a technology company fluctuated during the week within the neighborhood $E(24.5, 1.8)$ dollars. Determine the maximum and minimum price reached by the…
The share price of a technology company fluctuated during the week within the neighborhood $E(24.5, 1.8)$ dollars. Determine the maximum and minimum price reached by the shares and express it as a closed interval.
For the water in a swimming pool to be safe, its pH must be strictly in the interval $[7.2, 7.6]$. If we have an electronic meter that has a maximum absolute error of…
For the water in a swimming pool to be safe, its pH must be strictly in the interval $[7.2, 7.6]$. If we have an electronic meter that has a maximum absolute error of $0.1$, determine in which interval the device must give the reading so that we have absolute certainty that the real pH is safe. Express it also as a neighborhood.
Intervals are sets, so you can combine them. The union $A \cup B$ is everything in $A$ or $B$ (or both). The intersection $A \cap B$ is only what is in $A$ and $B$.
The best method: draw both intervals on stacked number lines, then read off the union (everything covered) and the intersection (covered by both). At each endpoint, ask whether that exact number is in the result.
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Given the intervals $A = [-4, 3)$ and $B = (0, 6]$, determine the resulting interval of the union $A \cup B$ and the intersection $A \cap B$.
Given the intervals $A = [-4, 3)$ and $B = (0, 6]$, determine the resulting interval of the union $A \cup B$ and the intersection $A \cap B$.
In a laboratory, the Alpha* strain of bacteria survives at temperatures in the interval $(-5, 20]$ ºF, while the Beta* strain survives in the interval $[10, 35)$ ºF.…
In a laboratory, the Alpha strain of bacteria survives at temperatures in the interval $(-5, 20]$ ºF, while the Beta strain survives in the interval $[10, 35)$ ºF. Determine the temperature interval in which both strains can coexist and express it as an inequality.
Many real questions have two conditions. "And" means the intersection of the two solution sets. "Or" means the union.
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A bridge is designed to support loads $C$ (in pounds) that satisfy the condition $|C - 4500| \le 1500$. On the other hand, a local regulation prohibits the passage of…
A bridge is designed to support loads $C$ (in pounds) that satisfy the condition $|C - 4500| \le 1500$. On the other hand, a local regulation prohibits the passage of vehicles exceeding $5500\text{ lb}$. Determine the real interval of loads that can cross the bridge satisfying both conditions.
Find the interval of real numbers $x$ that simultaneously satisfy the inequalities $3x - 2 \le 7$ and $5 - x < 4$. Once found, express it as a neighborhood.
Find the interval of real numbers $x$ that simultaneously satisfy the inequalities $3x - 2 \le 7$ and $5 - x < 4$. Once found, express it as a neighborhood.
The length of an interval with endpoints $a < b$ is $b - a$, whether or not the endpoints are included. This lets you compare pieces and compute percentages. To find what is left over when you remove one interval from another, subtract the overlap and keep the pieces that remain.
Astronomy problems use the same idea with distances measured in astronomical units (AU). One AU is the average distance from Earth to the Sun, about $93$ million miles, so Earth orbits at about $1$ AU. Some problems write this unit as UA. It is the same unit.
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A screw manufacturer accepts those whose length $L$ (in inches) satisfies the condition $|L - 30| < 0.5$. A batch of screws is received whose lengths vary uniformly in…
A screw manufacturer accepts those whose length $L$ (in inches) satisfies the condition $|L - 30| < 0.5$. A batch of screws is received whose lengths vary uniformly in the interval $[29.2, 30.8]$. What percentage of the length range of this batch will be rejected by quality control?
The "habitable zone" of a red dwarf star is defined by a neighborhood with center $0.8\text{ UA}$ and radius $0.2\text{ UA}$. An exoplanet has an elliptical orbit whose…
The "habitable zone" of a red dwarf star is defined by a neighborhood with center $0.8\text{ UA}$ and radius $0.2\text{ UA}$. An exoplanet has an elliptical orbit whose distance from the star varies in the interval $[0.5, 1.1]\text{ UA}$. Determine which intervals of the planet's orbit fall outside the habitable zone.