Algebra 1 · Unit 2
Write a solution set with set-builder notation.
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The bucket
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
{x | x < 4} reads “the set containing x such that x is less than 4.”
Two ways to write it
A ray can be written with infinity, or without it. Both buckets hold the same numbers.
{x | −∞ < x < 4}
{x | x < 4}
If both ends are numbers, write both ends in the bucket. There is no infinity to leave off.
Write both forms.
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
x < 4
{x | −∞ < x < 4}
{x | x < 4}
Both mean the same set.
Answer: {x | −∞ < x < 4}, which is the same as {x | x < 4}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
x > −1
{x | −1 < x < ∞}
{x | x > −1}
Both mean the same set.
Answer: {x | −1 < x < ∞}, which is the same as {x | x > −1}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
x ≤ 6
{x | −∞ < x ≤ 6}
{x | x ≤ 6}
Both mean the same set.
Answer: {x | −∞ < x ≤ 6}, which is the same as {x | x ≤ 6}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
x ≥ 0
{x | 0 ≤ x < ∞}
{x | x ≥ 0}
Both mean the same set.
Answer: {x | 0 ≤ x < ∞}, which is the same as {x | x ≥ 0}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
x < −3
{x | −∞ < x < −3}
{x | x < −3}
Both mean the same set.
Answer: {x | −∞ < x < −3}, which is the same as {x | x < −3}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
x ≥ 8
{x | 8 ≤ x < ∞}
{x | x ≥ 8}
Both mean the same set.
Answer: {x | 8 ≤ x < ∞}, which is the same as {x | x ≥ 8}
If both ends are numbers, write both ends.
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
−2 < x ≤ 5
Both ends are numbers, so both ends go in the bucket.
{x | −2 < x ≤ 5}
Answer: {x | −2 < x ≤ 5}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
−4 ≤ x < 1
Both ends are numbers, so both ends go in the bucket.
{x | −4 ≤ x < 1}
Answer: {x | −4 ≤ x < 1}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
2 ≤ x ≤ 7
Both ends are numbers, so both ends go in the bucket.
{x | 2 ≤ x ≤ 7}
Answer: {x | 2 ≤ x ≤ 7}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
{x | x > 5}
{x | 5 < x < ∞}
x > 5
Both mean the same set.
Answer: {x | 5 < x < ∞}, which is the same as x > 5
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
{x | −∞ < x ≤ −1}
{x | x ≤ −1}
x ≤ −1
Both mean the same set.
Answer: {x | x ≤ −1}, which is the same as x ≤ −1
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
−6 < x < 0
Both ends are numbers, so both ends go in the bucket.
{x | −6 < x < 0}
Answer: {x | −6 < x < 0}
Or stays inside one bucket. A bounded interval has one form.
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
x < −2 or x ≥ 3
{x | −∞ < x < −2 or 3 ≤ x < ∞}
{x | x < −2 or x ≥ 3}
Both mean the same set.
Answer: {x | −∞ < x < −2 or 3 ≤ x < ∞}, which is the same as {x | x < −2 or x ≥ 3}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
x ≤ 1 or x > 4
{x | −∞ < x ≤ 1 or 4 < x < ∞}
{x | x ≤ 1 or x > 4}
Both mean the same set.
Answer: {x | −∞ < x ≤ 1 or 4 < x < ∞}, which is the same as {x | x ≤ 1 or x > 4}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
The number line is x < 0 or x > 2.
{x | −∞ < x < 0 or 2 < x < ∞}
{x | x < 0 or x > 2}
Both mean the same set.
Answer: {x | −∞ < x < 0 or 2 < x < ∞}, which is the same as {x | x < 0 or x > 2}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
x ≥ −4
{x | −4 ≤ x < ∞}
{x | x ≥ −4}
Both mean the same set.
Answer: {x | −4 ≤ x < ∞}, which is the same as {x | x ≥ −4}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
−5 < x ≤ 2
Both ends are numbers, so both ends go in the bucket.
{x | −5 < x ≤ 2}
Answer: {x | −5 < x ≤ 2}
The braces { } mean the set containing. They are a bucket that holds the solution. The bar | reads such that.
Both ends go on forever.
{x | −∞ < x < ∞}
This bucket holds every real number.
Answer: {x | −∞ < x < ∞}
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