GCSE Maths: Inequalities on a number line (Higher)
Part of the GCSE Maths inequalities chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Inequalities on a number line
A number line is a way to see all the values an inequality allows. Imagine drawing a straight line with the numbers marked along it, then colouring every value that is a solution.
We need two pieces of information: where it starts or stops and which way it goes.
- The circle. We draw a circle on the boundary number. If the boundary number is a solution (the sign is ≤ or ≥), we fill the circle in: a closed circle. If the boundary number is not a solution (the sign is < or >), we leave the circle empty: an open circle.
- The arrow or line. We draw an arrow from the circle along the values that work. For x > 2 these are the bigger numbers, so the arrow points right. For x < 2 they are the smaller numbers, so the arrow points left.
Two ends. Some inequalities have a smallest and a largest value, for example −2 ≤ x < 3. This is read “x is bigger than or equal to −2 and smaller than 3”. We draw a circle at each end (closed at −2, open at 3) and join them with a line.
Whole numbers only. If the question says the letter is a whole number (an integer), the solutions are separate points, not a continuous line. We mark them with dots.
Key ideas
- Closed (filled) circle = the number is included (≤ or ≥).
- Open circle = the number is not included (< or >).
- The arrow points to the values that work: right for >, ≥; left for <, ≤.
- For a range such as 1 < x ≤ 5, draw a circle at each end and join them with a line.
- For whole-number solutions only, use separate dots.
Key words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.
Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.
- Look at the circle. The circle is open, so the number is not included. We need < or >, with no “or equal to”.Circle at 3: open
- Look at the arrow. An arrow pointing left shows values smaller than the circle; an arrow pointing right shows larger values.Arrow points left
- Write the inequality. Put the letter in the middle for a range, or on the left for a single end.x < 3
Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓
Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.
Key words. Closed circle (filled) = included. Open circle = not included.
Method. Mark the boundary with a circle, then draw an arrow (or line) over every value that works.
- Find the boundary number(s). These are the numbers where the inequality starts or stops.−1
- Choose open or closed circles. If the sign has the “or equal to” line (≤ or ≥) the number is included: closed (filled) circle. If it is < or > the number is not included: open circle.−1: closed
- Draw the line or arrow. Draw it over all the values that make the inequality true.a closed (filled) circle at −1 with an arrow pointing left
Check. Pick a number on your line and put it into x ≤ −1: it makes the inequality true. ✓
Watch out. Do not forget the circle. And an arrow that points the wrong way shows the wrong set of numbers.
Key words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.
Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.
- Look at the two circles. A closed circle means the number is included (≤). An open circle means it is not (<).Left circle at −2: closed
Right circle at 3: open - Look at the line between them. Every number on the line is a solution, so the letter is between the two circles.−2 and 3 are the two ends
- Write the inequality. Put the letter in the middle for a range, or on the left for a single end.−2 ≤ x < 3
Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓
Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.
Key words. Closed circle (filled) = included. Open circle = not included.
Method. Mark the boundary with a circle, then draw an arrow (or line) over every value that works.
- Find the boundary number(s). These are the numbers where the inequality starts or stops.1, 5
- Choose open or closed circles. If the sign has the “or equal to” line (≤ or ≥) the number is included: closed (filled) circle. If it is < or > the number is not included: open circle.1: open; 5: closed
- Draw the line or arrow. Draw it over all the values that make the inequality true.an open circle at 1, a closed (filled) circle at 5, and a line joining them
Check. Pick a number on your line and put it into 1 < x ≤ 5: it makes the inequality true. ✓
Watch out. Do not forget the circle. And an arrow that points the wrong way shows the wrong set of numbers.
Watch out
The most common mistake is the wrong circle. Open means “stop just before this number” (< or >). Closed means “include this number” (≤ or ≥). Also check the arrow: x < 3 points left towards the smaller numbers, even though the symbol itself looks like it opens to the right.Exam tip
Check any number-line answer by picking one number on the line and one number off it. Put both into the inequality: the first should work and the second should not.Exercise 10B Number lines
Non-calculatorKey words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.
Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.
- Look at the circle. The circle is open, so the number is not included. We need < or >, with no “or equal to”.Circle at 2: open
- Look at the arrow. An arrow pointing left shows values smaller than the circle; an arrow pointing right shows larger values.Arrow points left
- Write the inequality. Put the letter in the middle for a range, or on the left for a single end.x < 2
Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓
Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.
Key words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.
Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.
- Look at the circle. The circle is closed (filled), so the number is included. We need ≤ or ≥.Circle at −1: closed
- Look at the arrow. An arrow pointing left shows values smaller than the circle; an arrow pointing right shows larger values.Arrow points right
- Write the inequality. Put the letter in the middle for a range, or on the left for a single end.x ≥ −1
Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓
Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.
Key words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.
Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.
- Look at the circle. The circle is closed (filled), so the number is included. We need ≤ or ≥.Circle at −3: closed
- Look at the arrow. An arrow pointing left shows values smaller than the circle; an arrow pointing right shows larger values.Arrow points left
- Write the inequality. Put the letter in the middle for a range, or on the left for a single end.x ≤ −3
Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓
Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.
Key words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.
Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.
- Look at the circle. The circle is open, so the number is not included. We need < or >, with no “or equal to”.Circle at 0: open
- Look at the arrow. An arrow pointing left shows values smaller than the circle; an arrow pointing right shows larger values.Arrow points right
- Write the inequality. Put the letter in the middle for a range, or on the left for a single end.x > 0
Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓
Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.
Key words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.
Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.
- Look at the two circles. A closed circle means the number is included (≤). An open circle means it is not (<).Left circle at −2: closed
Right circle at 3: closed - Look at the line between them. Every number on the line is a solution, so the letter is between the two circles.−2 and 3 are the two ends
- Write the inequality. Put the letter in the middle for a range, or on the left for a single end.−2 ≤ x ≤ 3
Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓
Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.
Key words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.
Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.
- Look at the two circles. A closed circle means the number is included (≤). An open circle means it is not (<).Left circle at 1: open
Right circle at 6: closed - Look at the line between them. Every number on the line is a solution, so the letter is between the two circles.1 and 6 are the two ends
- Write the inequality. Put the letter in the middle for a range, or on the left for a single end.1 < x ≤ 6
Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓
Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.
Key words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.
Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.
- Look at the two circles. A closed circle means the number is included (≤). An open circle means it is not (<).Left circle at −4: open
Right circle at −1: open - Look at the line between them. Every number on the line is a solution, so the letter is between the two circles.−4 and −1 are the two ends
- Write the inequality. Put the letter in the middle for a range, or on the left for a single end.−4 < x < −1
Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓
Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.
Key words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.
Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.
- Look at the two circles. A closed circle means the number is included (≤). An open circle means it is not (<).Left circle at 0: closed
Right circle at 5: open - Look at the line between them. Every number on the line is a solution, so the letter is between the two circles.0 and 5 are the two ends
- Write the inequality. Put the letter in the middle for a range, or on the left for a single end.0 ≤ x < 5
Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓
Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.
Key words. Closed circle (filled) = included. Open circle = not included.
Method. Mark the boundary with a circle, then draw an arrow (or line) over every value that works.
- Find the boundary number(s). These are the numbers where the inequality starts or stops.3
- Choose open or closed circles. If the sign has the “or equal to” line (≤ or ≥) the number is included: closed (filled) circle. If it is < or > the number is not included: open circle.3: open
- Draw the line or arrow. Draw it over all the values that make the inequality true.an open circle at 3 with an arrow pointing right
Check. Pick a number on your line and put it into x > 3: it makes the inequality true. ✓
Watch out. Do not forget the circle. And an arrow that points the wrong way shows the wrong set of numbers.
Key words. Closed circle (filled) = included. Open circle = not included.
Method. Mark the boundary with a circle, then draw an arrow (or line) over every value that works.
- Find the boundary number(s). These are the numbers where the inequality starts or stops.−1
- Choose open or closed circles. If the sign has the “or equal to” line (≤ or ≥) the number is included: closed (filled) circle. If it is < or > the number is not included: open circle.−1: closed
- Draw the line or arrow. Draw it over all the values that make the inequality true.a closed (filled) circle at −1 with an arrow pointing left
Check. Pick a number on your line and put it into x ≤ −1: it makes the inequality true. ✓
Watch out. Do not forget the circle. And an arrow that points the wrong way shows the wrong set of numbers.
Key words. Closed circle (filled) = included. Open circle = not included.
Method. Mark the boundary with a circle, then draw an arrow (or line) over every value that works.
- Find the boundary number(s). These are the numbers where the inequality starts or stops.0
- Choose open or closed circles. If the sign has the “or equal to” line (≤ or ≥) the number is included: closed (filled) circle. If it is < or > the number is not included: open circle.0: open
- Draw the line or arrow. Draw it over all the values that make the inequality true.an open circle at 0 with an arrow pointing left
Check. Pick a number on your line and put it into x < 0: it makes the inequality true. ✓
Watch out. Do not forget the circle. And an arrow that points the wrong way shows the wrong set of numbers.
Key words. Closed circle (filled) = included. Open circle = not included.
Method. Mark the boundary with a circle, then draw an arrow (or line) over every value that works.
- Find the boundary number(s). These are the numbers where the inequality starts or stops.2
- Choose open or closed circles. If the sign has the “or equal to” line (≤ or ≥) the number is included: closed (filled) circle. If it is < or > the number is not included: open circle.2: closed
- Draw the line or arrow. Draw it over all the values that make the inequality true.a closed (filled) circle at 2 with an arrow pointing right
Check. Pick a number on your line and put it into x ≥ 2: it makes the inequality true. ✓
Watch out. Do not forget the circle. And an arrow that points the wrong way shows the wrong set of numbers.
Key words. Closed circle (filled) = included. Open circle = not included.
Method. Mark the boundary with a circle, then draw an arrow (or line) over every value that works.
- Find the boundary number(s). These are the numbers where the inequality starts or stops.−1, 3
- Choose open or closed circles. If the sign has the “or equal to” line (≤ or ≥) the number is included: closed (filled) circle. If it is < or > the number is not included: open circle.−1: open; 3: closed
- Draw the line or arrow. Draw it over all the values that make the inequality true.an open circle at −1, a closed (filled) circle at 3, and a line joining them
Check. Pick a number on your line and put it into −1 < x ≤ 3: it makes the inequality true. ✓
Watch out. Do not forget the circle. And an arrow that points the wrong way shows the wrong set of numbers.
Key words. Closed circle (filled) = included. Open circle = not included.
Method. Mark the boundary with a circle, then draw an arrow (or line) over every value that works.
- Find the boundary number(s). These are the numbers where the inequality starts or stops.2, 5
- Choose open or closed circles. If the sign has the “or equal to” line (≤ or ≥) the number is included: closed (filled) circle. If it is < or > the number is not included: open circle.2: closed; 5: open
- Draw the line or arrow. Draw it over all the values that make the inequality true.a closed (filled) circle at 2, an open circle at 5, and a line joining them
Check. Pick a number on your line and put it into 2 ≤ x < 5: it makes the inequality true. ✓
Watch out. Do not forget the circle. And an arrow that points the wrong way shows the wrong set of numbers.
Key words. A number line shows numbers in order. We use a circle at the boundary number and a line or arrow to show all the values that work.
Method. Read the circle (open or closed) and the direction of the arrow or line, then write the inequality.
- Look at the circle. The circle is closed (filled), so the number is included. We need ≤ or ≥.Circle at 3: closed
- Look at the arrow. An arrow pointing left shows values smaller than the circle; an arrow pointing right shows larger values.Arrow points left
- Write the inequality. Put the letter in the middle for a range, or on the left for a single end.x ≤ 3
Check. Pick a number on the line and substitute it. It should make your inequality true. Pick a number off the line: it should make it false. ✓
Watch out. An open circle means “not included”: use < or >. A closed circle means “included”: use ≤ or ≥.
Key words. A solution of an inequality is a number that makes it true.
Method. A number is a solution if the inequality is true when you put the number in place of x.
- Replace the letter with the number. We are testing whether 3 is one of the values that x can be.3 > 3 ?
- Compare the numbers. Decide if the statement is true.3 is equal to 3
- Answer. No: the statement is false, so 3 is not allowed.No
Check. On a number line, 3 is the boundary and 3 is on the wrong side of it. ✓
Watch out. With > the boundary number 3 itself is not included.
Key words. A solution of an inequality is a number that makes it true.
Method. A number is a solution if the inequality is true when you put the number in place of x.
- Replace the letter with the number. We are testing whether 3 is one of the values that x can be.3 ≥ 3 ?
- Compare the numbers. Decide if the statement is true.3 is equal to 3
- Answer. Yes: the statement is true, so 3 is allowed.Yes
Check. On a number line, 3 is the boundary and 3 is on the correct side of it. ✓
Watch out. With ≥ the boundary number 3 itself is included.
Key words. A solution of an inequality is a number that makes it true.
Method. A number is a solution if the inequality is true when you put the number in place of x.
- Replace the letter with the number. We are testing whether 3 is one of the values that x can be.3 < 3 ?
- Compare the numbers. Decide if the statement is true.3 is equal to 3
- Answer. No: the statement is false, so 3 is not allowed.No
Check. On a number line, 3 is the boundary and 3 is on the wrong side of it. ✓
Watch out. With < the boundary number 3 itself is not included.
Key words. A solution of an inequality is a number that makes it true.
Method. A number is a solution if the inequality is true when you put the number in place of x.
- Replace the letter with the number. We are testing whether 3 is one of the values that x can be.3 ≤ 3 ?
- Compare the numbers. Decide if the statement is true.3 is equal to 3
- Answer. Yes: the statement is true, so 3 is allowed.Yes
Check. On a number line, 3 is the boundary and 3 is on the correct side of it. ✓
Watch out. With ≤ the boundary number 3 itself is included.
Key words. A solution of an inequality is a number that makes it true.
Method. A number is a solution if the inequality is true when you put the number in place of x.
- Replace the letter with the number. We are testing whether −1 is one of the values that x can be.−1 > −2 ?
- Compare the numbers. Decide if the statement is true.−1 is bigger than −2
- Answer. Yes: the statement is true, so −1 is allowed.Yes
Check. On a number line, −2 is the boundary and −1 is on the correct side of it. ✓
Key words. A solution of an inequality is a number that makes it true.
Method. A number is a solution if the inequality is true when you put the number in place of x.
- Replace the letter with the number. We are testing whether 4 is one of the values that x can be.4 ≤ 4 ?
- Compare the numbers. Decide if the statement is true.4 is equal to 4
- Answer. Yes: the statement is true, so 4 is allowed.Yes
Check. On a number line, 4 is the boundary and 4 is on the correct side of it. ✓
Watch out. With ≤ the boundary number 4 itself is included.
Key words. A solution of an inequality is a number that makes it true.
Method. A number is a solution if the inequality is true when you put the number in place of x.
- Replace the letter with the number. We are testing whether 1 is one of the values that x can be.1 < 0.5 ?
- Compare the numbers. Decide if the statement is true.1 is bigger than 0.5
- Answer. No: the statement is false, so 1 is not allowed.No
Check. On a number line, 0.5 is the boundary and 1 is on the wrong side of it. ✓
Key words. A solution of an inequality is a number that makes it true.
Method. A number is a solution if the inequality is true when you put the number in place of x.
- Replace the letter with the number. We are testing whether −5 is one of the values that x can be.−5 ≥ −3 ?
- Compare the numbers. Decide if the statement is true.−5 is smaller than −3
- Answer. No: the statement is false, so −5 is not allowed.No
Check. On a number line, −3 is the boundary and −5 is on the wrong side of it. ✓
Key words. At least means this or more. At most means this or less. A closed circle means included, an open circle means not included.
Method. Spot the key phrase, write the inequality, then mark the boundary with a circle and draw the arrow.
- Find the key phrase. The phrase tells us which symbol to use.“at least”
- Decide if the number itself is allowed. This phrase allows the boundary number itself, so we need ≤ or ≥.≥
- Write the inequality. x stands for the number.x ≥ −2
- Draw it on a number line. Closed (filled) circle because the number is included. The arrow points right because the values are bigger.a closed (filled) circle at −2 with an arrow pointing right
Check. Pick a number on the arrow and put it in the inequality: it makes it true. ✓
Watch out. Open circle = number not included (<, >). Closed circle = included (≤, ≥).
Key words. At least means this or more. At most means this or less. A closed circle means included, an open circle means not included.
Method. Spot the key phrase, write the inequality, then mark the boundary with a circle and draw the arrow.
- Find the key phrase. The phrase tells us which symbol to use.“less than”
- Decide if the number itself is allowed. This phrase does not allow the number itself, so we need < or >.<
- Write the inequality. x stands for the number.x < 4
- Draw it on a number line. Open circle because the number is not included. The arrow points left because the values are smaller.an open circle at 4 with an arrow pointing left
Check. Pick a number on the arrow and put it in the inequality: it makes it true. ✓
Watch out. Open circle = number not included (<, >). Closed circle = included (≤, ≥).
Key words. At least means this or more. At most means this or less. A closed circle means included, an open circle means not included.
Method. Spot the key phrase, write the inequality, then mark the boundary with a circle and draw the arrow.
- Find the key phrase. The phrase tells us which symbol to use.“more than”
- Decide if the number itself is allowed. This phrase does not allow the number itself, so we need < or >.>
- Write the inequality. x stands for the number.x > 0
- Draw it on a number line. Open circle because the number is not included. The arrow points right because the values are bigger.an open circle at 0 with an arrow pointing right
Check. Pick a number on the arrow and put it in the inequality: it makes it true. ✓
Watch out. Open circle = number not included (<, >). Closed circle = included (≤, ≥).
Key words. At least means this or more. At most means this or less. A closed circle means included, an open circle means not included.
Method. Spot the key phrase, write the inequality, then mark the boundary with a circle and draw the arrow.
- Find the key phrase. The phrase tells us which symbol to use.“at most”
- Decide if the number itself is allowed. This phrase allows the boundary number itself, so we need ≤ or ≥.≤
- Write the inequality. x stands for the number.x ≤ 6
- Draw it on a number line. Closed (filled) circle because the number is included. The arrow points left because the values are smaller.a closed (filled) circle at 6 with an arrow pointing left
Check. Pick a number on the arrow and put it in the inequality: it makes it true. ✓
Watch out. Open circle = number not included (<, >). Closed circle = included (≤, ≥).
Key words. An integer is a whole number (positive, negative or zero).
Method. Mark each allowed whole number, then write the smallest and largest as the ends of the inequality.
- Decide what values are allowed. n is a whole number, so only the integers from 0 to 8 are possible.0, 1, 2, 3, 4, 5, 6, 7, 8
- Mark each whole number with a dot. Because n must be a whole number, we put separate dots rather than a continuous line.
- Write the inequality. The smallest value is included and so is the largest, so we use ≤ both times.0 ≤ n ≤ 8
Check. Count the integers: from 0 to 8 there are 9 whole numbers. ✓
Watch out. For whole numbers use separate dots, not a solid line, because values like 2.5 are not allowed.
Method. “At least 110” means ≥ 110, and “less than 195” means < 195. Join them into one inequality.
- Write the lower limit. “At least 110” includes 110.h ≥ 110
- Write the upper limit. “Less than 195” does not include 195.h < 195
- Join them. Put h in the middle, with the smaller number on the left.110 ≤ h < 195
- Draw the number line. Closed circle at 110, open circle at 195, line between.
Check. Test h = 110: allowed (closed circle). Test h = 195: not allowed (open circle). ✓
Watch out. Check which end gets the closed circle: the one with ≤ or ≥.
Method. Look carefully at the circle on the number line.
- Look at the circle. It is filled in, which is a closed circle. A closed circle means the number 4 is included.Closed circle at 4
- Match to a symbol. < does not include 4, so it needs an open circle. A closed circle needs ≤.≤
- Write the correct inequality. The arrow points left, so the values are smaller.x ≤ 4
Check. 4 is on the line (closed circle), so 4 must be allowed. 4 < 4 is false but 4 ≤ 4 is true. ✓
Watch out. Open circle = < or >. Closed circle = ≤ or ≥.
Method. Compare what each symbol allows, then how that is drawn.
- Say what x > 3 allows. It allows numbers bigger than 3, such as 3.1 and 4, but not 3 itself.Open circle at 3, arrow pointing right
- Say what x ≥ 3 allows. It allows numbers bigger than 3 and 3 itself.Closed circle at 3, arrow pointing right
- State the difference. The only difference is whether 3 is a solution.x ≥ 3 includes 3; x > 3 does not.
Check. Test x = 3: 3 > 3 is false but 3 ≥ 3 is true. ✓
Watch out. The two arrows point the same way; only the circle changes.
Method. Read the dots from left to right, then use the smallest and the largest as the ends of the inequality.
- Read the dots. The dots sit on whole numbers.−2, −1, 0, 1
- Find the smallest and largest. They are the two ends of the range.smallest −2, largest 1
- Write the inequality with ≤ twice. Both end values are allowed.−2 ≤ n ≤ 1
Check. List the whole numbers that satisfy it: −2, −1, 0, 1 — matches the dots. ✓
Watch out. This is only one correct answer. −3 < n < 2 also gives the same whole numbers.
Method. There are two separate arrows, so there are two inequalities joined by “or”.
- Read the left arrow. Closed circle at −1, pointing left.x ≤ −1
- Read the right arrow. Open circle at 3, pointing right.x > 3
- Join with “or”. A value works if it is in either part.x ≤ −1 or x > 3
Check. Test x = 0: it is between the two arrows, so it should not work. 0 ≤ −1 and 0 > 3 are both false. ✓
Watch out. Never write this as a single inequality like 3 < x ≤ −1; it does not make sense.
Method. The values that satisfy both are where the two arrows overlap.
- Find the lower limit. The first inequality says x is bigger than −1 (open circle).x > −1
- Find the upper limit. The second says x is at most 4 (closed circle).x ≤ 4
- Join them. Put the letter in the middle, smaller number on the left.−1 < x ≤ 4
Check. Test x = 2: 2 > −1 and 2 ≤ 4, both true. Test x = 5: 5 ≤ 4 is false. ✓
Watch out. Keep each circle type: the −1 stays open and the 4 stays closed.