Inequalities with Variables on Both Sides
When an inequality like has variable terms on both sides, the first job is to gather them onto one side. Add or subtract a variable term from both sides until only one side has an — then it's a two-step inequality you already know how to finish.
Here's the part students like: adding or subtracting variable terms never flips the inequality sign. The only move that ever flips it is multiplying or dividing both sides by a negative number, and with one smart choice at the start, you can usually avoid that entirely.
Gather the variables on one side
Pick one side to hold the variable and subtract the other side's variable term from both sides. In , subtracting from both sides leaves — one variable side, and the symbol never moved.
From there it's the standard finish: subtract from both sides to get , then divide by to get .
Choose the side that keeps the coefficient positive
You can collect the variables on either side — the solution comes out the same. But one choice is cleaner: move the smaller variable term over to the larger one, so the remaining coefficient is positive and no flip is ever needed.
Take . Subtracting from both sides gives , which works but forces a divide-by-negative-and-flip later. Subtracting instead gives — positive coefficient, no flip coming. Then , so , which is the same statement as .
If you do end up dividing by a negative, that's fine — just remember the flip. Both paths land on the same answer.
Reading an answer like
When the variable lands on the right, flip your reading, not the math: says is less than or equal to , which means is greater than or equal to , or . Notice the symbol still opens toward the in both versions — the open end of the symbol always faces the larger side.
Worked examples
Example 1: collect on the left
Solve .
Answer:
Example 2: collect on the right to avoid a flip
Solve .
Answer:
Example 3: variable term with a negative sign
Solve .
Answer:
Try one yourself
Common questions
Does moving a variable term across the inequality flip the sign?
No. Moving a variable term is adding or subtracting the same thing from both sides, and adding or subtracting never flips the symbol. Only multiplying or dividing both sides by a negative number does.
Which side should I collect the variables on?
Either side gives the same solution, but collecting them where the coefficient stays positive is cleaner. Move the smaller variable term to the side with the larger one and you'll never be forced to divide by a negative.
What if the variables cancel completely?
Then the answer is about the leftover number statement. If you reach something always true, like , every real number is a solution. If you reach something false, like , there is no solution.
How do I turn into an -first answer?
Swap the sides and reverse the symbol: becomes . The relationship hasn't changed — the open end of the symbol still faces the , the larger side — you've just written the same fact with first.
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