PCAL- Interval Notation worksheet - iklflm £3 - 81 Mod ... - Free Printable

The image you uploaded appears to be a worksheet focused on interval notation. Interval notation is a way of describing subsets of real numbers using intervals. The task involves translating inequalities or conditions into interval notation.

Below, I will solve each problem step by step and explain the solution.

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Problem 1:

Inequality: \( x \geq -3 \)
Solution:
- The inequality \( x \geq -3 \) means that \( x \) can take any value greater than or equal to \(-3\).
- In interval notation, this is written as:
\[
[-3, \infty)
\]
- The square bracket \([\) indicates that \(-3\) is included.
- The parenthesis \((\) indicates that the interval extends to infinity without including it.

Answer: \([-3, \infty)\)

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Problem 2:

Inequality: \( x < 5 \)
Solution:
- The inequality \( x < 5 \) means that \( x \) can take any value less than \(5\).
- In interval notation, this is written as:
\[
(-\infty, 5)
\]
- The parenthesis \((\) indicates that the interval extends to negative infinity without including it.
- The parenthesis \((\) also indicates that \(5\) is not included.

Answer: \((-∞, 5)\)

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Problem 3:

Inequality: \( x > -2 \)
Solution:
- The inequality \( x > -2 \) means that \( x \) can take any value greater than \(-2\).
- In interval notation, this is written as:
\[
(-2, \infty)
\]
- The parenthesis \((\) indicates that \(-2\) is not included.
- The parenthesis \((\) also indicates that the interval extends to infinity without including it.

Answer: \((-2, ∞)\)

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Problem 4:

Inequality: \( x \leq 7 \)
Solution:
- The inequality \( x \leq 7 \) means that \( x \) can take any value less than or equal to \(7\).
- In interval notation, this is written as:
\[
(-\infty, 7]
\]
- The parenthesis \((\) indicates that the interval extends to negative infinity without including it.
- The square bracket \(\]\) indicates that \(7\) is included.

Answer: \((-∞, 7]\)

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Problem 5:

Inequality: \( -4 \leq x \leq 6 \)
Solution:
- The inequality \( -4 \leq x \leq 6 \) means that \( x \) can take any value between \(-4\) and \(6\), inclusive.
- In interval notation, this is written as:
\[
[-4, 6]
\]
- The square brackets \([\) and \(\]\) indicate that both \(-4\) and \(6\) are included.

Answer: \([-4, 6]\)

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Problem 6:

Inequality: \( -1 < x < 8 \)
Solution:
- The inequality \( -1 < x < 8 \) means that \( x \) can take any value between \(-1\) and \(8\), exclusive.
- In interval notation, this is written as:
\[
(-1, 8)
\]
- The parentheses \((\) and \((\) indicate that neither \(-1\) nor \(8\) is included.

Answer: \((-1, 8)\)

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Problem 7:

Inequality: \( x \leq -5 \) or \( x > 2 \)
Solution:
- This is a compound inequality with two parts:
1. \( x \leq -5 \): This means \( x \) can be any value less than or equal to \(-5\).
2. \( x > 2 \): This means \( x \) can be any value greater than \(2\).
- In interval notation, we write these as separate intervals and use the union symbol (\(\cup\)) to combine them:
\[
(-\infty, -5] \cup (2, \infty)
\]
- The first interval \((-∞, -5]\) includes \(-5\) but not \(-∞\).
- The second interval \((2, ∞)\) does not include \(2\) but extends to \(∞\).

Answer: \((-∞, -5] \cup (2, ∞)\)

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Problem 8:

Inequality: \( x < -3 \) or \( x \geq 1 \)
Solution:
- This is another compound inequality with two parts:
1. \( x < -3 \): This means \( x \) can be any value less than \(-3\).
2. \( x \geq 1 \): This means \( x \) can be any value greater than or equal to \(1\).
- In interval notation, we write these as separate intervals and use the union symbol (\(\cup\)) to combine them:
\[
(-\infty, -3) \cup [1, \infty)
\]
- The first interval \((-∞, -3)\) does not include \(-3\) but extends to \(-∞\).
- The second interval \([1, ∞)\) includes \(1\) but extends to \(∞\).

Answer: \((-∞, -3) \cup [1, ∞)\)

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Problem 9:

Condition: \( x \) is any real number
Solution:
- If \( x \) is any real number, then \( x \) can take any value from negative infinity to positive infinity.
- In interval notation, this is written as:
\[
(-\infty, \infty)
\]
- The parentheses \((\) and \((\) indicate that the interval extends infinitely in both directions.

Answer: \((-∞, ∞)\)

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Problem 10:

Condition: \( x \) is any real number except \( x = 4 \)
Solution:
- If \( x \) is any real number except \( x = 4 \), then \( x \) can take any value except \(4\).
- In interval notation, we write this as two separate intervals combined with the union symbol (\(\cup\)):
\[
(-\infty, 4) \cup (4, \infty)
\]
- The first interval \((-∞, 4)\) includes all values less than \(4\).
- The second interval \((4, ∞)\) includes all values greater than \(4\).
- The value \(4\) is excluded in both intervals.

Answer: \((-∞, 4) \cup (4, ∞)\)

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Final Answers:

1. \([-3, \infty)\)
2. \((-∞, 5)\)
3. \((-2, ∞)\)
4. \((-∞, 7]\)
5. \([-4, 6]\)
6. \((-1, 8)\)
7. \((-∞, -5] \cup (2, ∞)\)
8. \((-∞, -3) \cup [1, ∞)\)
9. \((-∞, ∞)\)
10. \((-∞, 4) \cup (4, ∞)\)

Boxed Final Answer:
\[
\boxed{
\begin{aligned}
1. & \; [-3, \infty) \\
2. & \; (-\infty, 5) \\
3. & \; (-2, \infty) \\
4. & \; (-\infty, 7] \\
5. & \; [-4, 6] \\
6. & \; (-1, 8) \\
7. & \; (-\infty, -5] \cup (2, \infty) \\
8. & \; (-\infty, -3) \cup [1, \infty) \\
9. & \; (-\infty, \infty) \\
10. & \; (-\infty, 4) \cup (4, \infty)
\end{aligned}
}
\]