Inequalities on a Number Line Worksheet | Printable PDF Worksheets - Free Printable

Here are the solutions to the problems on the worksheet, broken down by section.

Section A: Write down the inequality

*Rule:* An open circle ($\circ$) means $<$ or $>$. A closed circle ($\bullet$) means $\leq$ or $\geq$. The arrow points in the direction of the numbers that work.

1. Top Left: Open circle at $-1$, arrow to the right.
* Answer: $x > -1$
2. Top Middle: Open circle at $-1$, arrow to the left.
* Answer: $x < -1$
3. Top Right: Open circle at $0$, arrow to the right.
* Answer: $x > 0$
4. Middle Left: Closed circle at $1$, arrow to the right.
* Answer: $x \geq 1$
5. Middle Center: Closed circle at $-2$, arrow to the right.
* Answer: $x \geq -2$
6. Middle Right: Closed circle at $-2$, arrow to the left.
* Answer: $x \leq -2$
7. Bottom Left: Open circle at $-1$, open circle at $3$, line between them.
* Answer: $-1 < x < 3$
8. Bottom Center: Closed circle at $-2$, closed circle at $1$, line between them.
* Answer: $-2 \leq x \leq 1$
9. Bottom Right: Open circle at $-3$, closed circle at $1$, line between them.
* Answer: $-3 < x \leq 1$

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Section B: Illustrate each inequality

*Rule:* Use an open circle for $<$ or $>$ and a closed circle for $\leq$ or $\geq$. Shade the line in the correct direction.

1. $x \leq -3$: Put a closed circle on $-3$ and shade to the left.
2. $x > 1$: Put an open circle on $1$ and shade to the right.
3. $x \geq -5$: Put a closed circle on $-5$ and shade to the right.
4. $-5 \leq x < 7.5$: Put a closed circle on $-5$ and an open circle on $7.5$ (halfway between 7 and 8). Connect them with a line.
5. $-6 < x \leq -2$: Put an open circle on $-6$ and a closed circle on $-2$. Connect them with a line.
6. $-2.5 \leq x \leq -0.5$: Put a closed circle on $-2.5$ (between -2 and -3) and a closed circle on $-0.5$ (between 0 and -1). Connect them.
7. $x \leq -3$: Same as #1. Closed circle on $-3$, shade left.
8. $-20 \leq x < 80$: This range is very wide. Put a closed circle on $-20$ and an open circle on $80$. Connect them.
9. $-0.5 < x < 0.7$: Put an open circle on $-0.5$ and an open circle on $0.7$. Connect them.

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Extension: Whole and Prime Numbers

Left Table: Whole Numbers
*(Whole numbers are $0, 1, 2, 3...$)*

1. $-1 \leq x < 3$:
* Numbers greater than or equal to $-1$ but less than $3$.
* Integers: $-1, 0, 1, 2$.
* Whole numbers only: $0, 1, 2$
2. $-22 \leq x \leq -19$:
* These are all negative numbers. There are no whole numbers here.
* Answer: None (or empty set)
3. $99 < x < 101$:
* The only integer strictly between 99 and 101 is 100.
* Answer: $100$
4. $0.8 < x < 6.9$:
* Integers inside this range: $1, 2, 3, 4, 5, 6$.
* Answer: $1, 2, 3, 4, 5, 6$

Right Table: Prime Numbers
*(Prime numbers are numbers divisible only by 1 and themselves, e.g., 2, 3, 5, 7... Note: 1 is NOT prime.)*

1. $2 \leq x < 13$:
* Integers: $2, 3, ..., 12$.
* Primes: $2, 3, 5, 7, 11$
2. $41 < x < 53$:
* Integers: $42, ..., 52$.
* Check for primes: 43 (yes), 47 (yes), 53 is not included because it is $< 53$.
* Answer: $43, 47$
3. $0 < 2x < 14$:
* First, divide everything by 2 to isolate $x$: $0 < x < 7$.
* Integers: $1, 2, 3, 4, 5, 6$.
* Primes in this list: $2, 3, 5$
4. $-22 \leq 1 - x < -16$:
* Subtract 1 from all parts: $-23 \leq -x < -17$.
* Divide by $-1$ and flip the signs: $23 \geq x > 17$ (which is the same as $17 < x \leq 23$).
* Integers: $18, 19, 20, 21, 22, 23$.
* Primes in this list: 19 and 23.
* Answer: $19, 23$

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Final Answer:

Section A
1. $x > -1$
2. $x < -1$
3. $x > 0$
4. $x \geq 1$
5. $x \geq -2$
6. $x \leq -2$
7. $-1 < x < 3$
8. $-2 \leq x \leq 1$
9. $-3 < x \leq 1$

Section B
*(Descriptions for drawing)*
1. Closed circle at -3, arrow left.
2. Open circle at 1, arrow right.
3. Closed circle at -5, arrow right.
4. Closed circle at -5, open circle at 7.5, line connecting them.
5. Open circle at -6, closed circle at -2, line connecting them.
6. Closed circle at -2.5, closed circle at -0.5, line connecting them.
7. Closed circle at -3, arrow left.
8. Closed circle at -20, open circle at 80, line connecting them.
9. Open circle at -0.5, open circle at 0.7, line connecting them.

Extension
*Whole Numbers:*
1. $0, 1, 2$
2. None
3. $100$
4. $1, 2, 3, 4, 5, 6$

*Prime Numbers:*
1. $2, 3, 5, 7, 11$
2. $43, 47$
3. $2, 3, 5$
4. $19, 23$