Inequalities and why the sign flips

TL;DR:

• Treat an inequality like an equation, except when you multiply or divide by a negative number: then the sign flips.

• For −2x > 6, dividing by −2 gives x < −3.

• Check with a test number, and show the answer on a number line or in interval notation.

Solving 2x > 6 is just like solving an equation: divide by 2 and get x > 3. But solve −2x > 6 the same way and the sign must turn around: x < −3. This is the step many students get wrong, and it is easy to understand once you try a few numbers.

What an inequality asks

An equation like x = 3 is true for one number. An inequality like x > 3 is true for a whole range of numbers: 3.1, 4, 100 and so on. The solution is a set, which you can picture on a number line or write with intervals.

Why multiplying by a negative flips the sign

Start with a true statement: 1 < 2. Multiply both sides by −1. The numbers become −1 and −2. On the number line −1 is to the right of −2, so −1 > −2. The order reversed.

Another example: 2 < 5. Multiply by −3: the results are −6 and −15, and −6 > −15. Multiplying by a negative reflects the number line through zero, which swaps left and right. Adding or subtracting any number, positive or negative, slides the line without reflecting it, so those steps never flip the sign.

Worked example: −2x > 6 gives x < −3

1. Divide both sides by −2.

2. Because −2 is negative, flip the sign: x < 6 / (−2).

3. Simplify: x < −3.

Check with test numbers. Choose x = −4, which is less than −3: −2(−4) = 8, and 8 > 6 is true. Choose x = −2, which is greater than −3: −2(−2) = 4, and 4 > 6 is false. Choose the boundary x = −3: −2(−3) = 6, and 6 > 6 is false, so −3 itself is not included.

Showing the answer

• Number line. Draw an open circle at −3 (the boundary is not included) and shade everything to the left.

• Interval notation. (−∞, −3). A round bracket means "not included". Infinity always gets a round bracket.

• Set notation. {x : x < −3}.

For a "less than or equal" sign, the boundary is included. Draw a filled circle and use a square bracket, such as (−∞, −3].

A longer example: 5 − 3x ≥ 14

1. Subtract 5 from both sides: −3x ≥ 9. (No flip, we only subtracted.)

2. Divide by −3 and flip: x ≤ −3.

Check the boundary: x = −3 gives 5 − 3(−3) = 5 + 9 = 14, and 14 ≥ 14 is true. Check a point inside the answer, x = −4: 5 + 12 = 17 ≥ 14, true. Check a point outside, x = 0: 5 ≥ 14, false. In interval notation, the answer is (−∞, −3].

Compound inequalities

Solve −3 < 2x + 1 ≤ 9 by doing the same step to all three parts. Subtract 1: −4 < 2x ≤ 8. Divide by 2 (positive, so no flip): −2 < x ≤ 4. The interval is (−2, 4]. Check x = 0: 2(0) + 1 = 1, which is between −3 and 9. Check x = 4: 2(4) + 1 = 9, and 9 ≤ 9 is allowed. Check x = −2: −3, and −3 < −3 is false, so −2 is excluded.

Try two more yourself

Solve −x/4 < 2. Multiply both sides by −4 and flip the sign: x > −8. Check x = 0: 0 < 2 is true, and 0 is greater than −8. Check x = −10: −(−10)/4 = 2.5 < 2 is false, and −10 is less than −8. So the answer is the interval (−8, ∞).

A word version: a phone plan costs 15 + 4g dollars for g gigabytes (the plan from the slope article), and your budget is $51. Then 15 + 4g ≤ 51, so 4g ≤ 36 and g ≤ 9. No sign flip is needed, since we divided by a positive 4. Check g = 9: 15 + 36 = 51, which is allowed. Check g = 10: 55, over budget. In context, g also cannot be negative, so the useful answer is 0 ≤ g ≤ 9.

A quick way to remember the rule is the number-line picture. Multiplying by a negative number is a reflection through zero, and a reflection swaps left and right, so the order of any two numbers swaps too. Positive numbers only stretch or shrink the line, so they never reverse the order. If you are unsure, try the step on a pair of numbers you can compare, such as 1 and 2, before applying it to an unknown.

How to check your answer

• Test number inside. It should make the original true.

• Test number outside. It should make the original false.

• Boundary. Decide whether it belongs by checking if the sign includes equality.

• Look at each step. Only multiplication or division by a negative should flip the sign.

Common mistakes

• Forgetting to flip when dividing by a negative.

• Flipping the sign when subtracting a negative number or moving a term. Only multiplication and division by a negative flip it.

• Including the boundary for a strict inequality.

• Writing a square bracket next to infinity.

• Reversing the sign twice by accident.

Where you will use it next

Inequalities describe regions and ranges. The distance form appears in solving absolute value equations, and a boundary line for a two-variable inequality comes from graphing a line from its equation.

Frequently asked questions

What if I multiply by a variable?

Be careful. If the variable might be negative, you do not know whether to flip the sign, so you must split into cases or avoid multiplying by it.

Why is the interval for x < −3 written (−∞, −3)?

Because the solutions run from negative infinity up to, but not including, −3.