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This page sets out the method for inequalities in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Inequalities topic page.
When you see a single inequality with on one side, undo each operation in turn.
When is trapped between two numbers, do every step to all three parts.
< means less than. > means greater than.
The extra bar on ≤ and ≥ means the end value is included.
In 2x + 1 < 9, subtract 1 and then divide by 2.
The sign stays facing the same way because you divided by a positive number.
If you multiply or divide both sides by a negative number, reverse the sign.
−2x < 6 becomes x > −3.
x < 3 is an open circle at 3 and a line to the left.
x ≥ 3 is a filled circle and a line to the right.
If x < 4 and x is a whole number, the largest possible x is 3.
Read whether the question wants the inequality or the list of integers.
Try a number on the side you shaded.
It should make the original inequality true. A number on the other side should not.
Treat x+6>15 as you would treat x+6=15, and keep the > sign. Subtract 6: x>9. The solution is every number bigger than 9, not the single number 9. On a number line the circle at 9 is empty and the arrow points right. A filled circle would mean x≥ 9, which includes a value the inequality has ruled out. Copy the symbol onto every line you write, including the answer line.
With a coefficient, undo the addition first. 3x-4≤ 17 becomes 3x≤ 21, then x≤ 7. Dividing by a positive number does not flip the symbol. Dividing by a negative number does flip it. The safer route, when the letter has a negative coefficient, is to add that term to the other side so the letter stays positive. Then you never flip, and you cannot forget to.
Method: 1. Solve x+6>15 with the same balancing steps as the equation x+6=15, and keep the > sign. 2. The result is every number greater than 9, not the single value 9. 3. On a number line, draw an empty circle at 9 and point the arrow to the right.
A double inequality such as -3<2x+1≤ 9 is three parts linked by two symbols. Do the same operation to all three parts. Subtract 1: -4<2x≤ 8. Divide by 2: -2<x≤ 4. The left symbol stays strict and the right symbol stays "or equal". Mixing them up changes which end is included.
The integers that satisfy -2<x≤ 4 are the whole numbers strictly above -2 and up to 4 inclusive. -2 itself is out. -1, 0, 1, 2, 3 and 4 are in. People drop 0 because it "does not look like a number", and they drop the negatives because they start counting at 1. Write the list from smallest to largest so a gap is obvious. Test one value inside and one value outside in the original inequality if the list feels uncertain.
Method: 1. Read -3<2x+1≤ 9 as three parts joined by two inequality signs. 2. Do each operation to the left part, the middle part and the right part. 3. Subtract 1 from all three parts to reach -4<2x≤ 8.
An arrow to the right means "and every number beyond". An arrow to the left means the same towards the negatives. A bar between two circles means "between these values". The circle, not the direction of your pen, decides whether the endpoint counts. Empty for < and >. Filled for ≤ and ≥.
A check uses a number, not a feeling. For x>5, try x=6 in the original inequality. It should make a true statement. Try x=5 as well: it should fail if the symbol is strict, and succeed if the symbol is ≥. This catches a flipped sign and a filled circle drawn for an empty one. It takes less time than reworking the whole question. On a double inequality, test both ends and one integer in the middle. An end that was supposed to be excluded must fail. An end that was supposed to be included must hold with equality.
Method: 1. A right-hand arrow means every number larger than the circled value. The circle says whether that value counts. 2. A left-hand arrow means every number smaller than the circled value. The circle says whether that value counts. 3. A bar joining two circles means the numbers that lie between those ends.
An inequality is solved like an equation, with one important exception. When you multiply or divide both sides by a negative number, the inequality sign must be reversed. In -3xge12 dividing by -3 gives x≤-4, not x≥-4. The reason is that negating numbers reverses their order: 2<5 but -2>-5.
You can avoid the flip altogether. Move the x term to the side where it is positive. To solve 4-x>1 add x to both sides to get 4>1+x, then subtract 1 to get 3>x, which is the same as x<3. Both routes give the same answer. Test with a number to be sure: x=0 gives 4>1, which is true, and x=5 gives -1>1, which is false.
Method: 1. Treat the inequality like an equation and undo additions and subtractions first. 2. If you divide or multiply by a negative number, reverse the sign. 3. Or move the x term across so that it stays positive and read the answer the other way round. 4. Test one value inside your solution and one outside it in the original inequality.
Tip: Write 3>x as x<3 at the end. The sign always points at the smaller number.
The method is the same as for an equation with letters on both sides. Collect the x terms on one side and the numbers on the other, then divide. To solve 7x+4ge3x-12 subtract 3x to get 4x+4≥-12, subtract 4 to get 4x≥-16 and divide by 4 to get x≥-4. The inequality sign stays the same because you divided by a positive number.
Fractions are handled by multiplying both sides by the denominator, which is positive so the sign does not change. For (3x-1)/2>x+4 multiply by 2 to get 3x-1>2x+8, so x>9. Leave your answer as an inequality such as x>9 and not as an equals sign. Marks are lost when students write x=9 at the end.
Method: 1. Subtract the smaller x term from both sides. 2. Add or subtract numbers so the plain numbers are on the other side. 3. Divide by the coefficient of x and keep the sign unless it is negative. 4. Write the final answer in the form x> or x≤ a number.
A double inequality such as -7le2x-3<9 has three parts and you do the same thing to all three. Add 3 to get -4le2x<12 and divide by 2 to get -2≤ x<6. Now list the whole numbers: -2, -1, 0, 1, 2, 3, 4, 5. The value -2 is included because the sign is ≤, and the value 6 is excluded because the sign is <.
Check the end points carefully every time. A common error is to forget the negative integers or to include a number that the strict sign rules out. If the inequality is 6le2x<15, the solution is 3≤ x<7.5, so the integers are 3, 4, 5, 6, 7. The end value 7.5 is not an integer, so you stop at the last whole number below it.
Method: 1. Apply each operation to all three parts of the double inequality. 2. Write the solution as a≤ x<b or similar. 3. List every integer from the smallest to the largest allowed, including negatives and zero. 4. Check the first and last integers against the original inequality.
Tip: Count the integers in your list. From -2 to 5 there are eight, not seven.
Exam questions use wording instead of symbols. "At most" and "no more than" mean ≤, "at least" and "not less than" mean ≥, "fewer than" and "less than" mean <, and "more than" means >. A number line shows the same information: an open circle means the end value is not included and a closed circle means it is. A line from an open circle at -2 to a closed circle at 3 is -2<xle3.
In a context question form the inequality first and solve it second. If a sandwich costs £4 and you have £30, then 4mle30 gives mle7.5. Because you cannot buy half a sandwich the greatest whole number is 7. Check by working out 4×7=28, which is within budget, and 4×8=32, which is not. Always answer the question that was asked, not just the inequality.
Method: 1. Translate the key words into a symbol: at most, at least, fewer than, more than. 2. Write an inequality using a letter for the unknown quantity. 3. Solve it and then apply any real-world rule, such as only whole numbers. 4. Test the boundary and the next whole number to confirm the answer.
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