Math · College algebra · Worked example
Solve a linear inequality with a sign flip
Solve 7 − 2x > 3x − 8. Write the solution in interval notation and check it.
Collect the x terms on one side
Subtract 3x from both sides. Subtracting never changes the direction of the sign, whatever is subtracted.
Collect the constants
Subtract 7 from both sides: −8 − 7 = −15.
Divide by −5 and reverse the sign
Dividing both sides by a negative number reverses their order, so > becomes <. Then −15 ÷ (−5) = 3.
Write the interval
Every number less than 3 works. The number 3 itself does not, because the original sign is strict, so both ends of the interval take parentheses.
Check with test values
Inside the interval, x = 0 gives 7 > −8, which is true. Outside it, x = 4 gives −1 > 4, which is false. At the endpoint, x = 3 gives 1 > 1, which is false, so 3 is rightly left out.
| x | 7 − 2x | 3x − 8 | True? |
|---|---|---|---|
| 0 | 7 | −8 | Yes |
| 3 | 1 | 1 | No (equal) |
| 4 | −1 | 4 | No |
Result
x < 3, which is (−∞, 3) in interval notation.
Your turn
Solve −4 ≤ 2x + 6 < 10 and write the answer in interval notation.
Show the answer and explanation
−5 ≤ x < 2, which is [−5, 2).
Subtract 6 from all three parts: −10 ≤ 2x < 4. Divide all three parts by 2, a positive number, so the signs stay: −5 ≤ x < 2. The bracket includes −5; the parenthesis leaves out 2.
Keep exploring
Open the steps in Math and change > to ≥ in the first line: every later line has to change with it, and the interval becomes (−∞, 3].
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Check the sign flip in Steps, assumptions & inequalities Check each step in Math Open worked example on a board Inequality rules in Math ReferenceYour existing work stays on this device. Examples open as editable copies.