Solving linear equations and inequalities

Solving a linear equation means undoing whatever was done to the variable, in reverse order, doing the same thing to both sides each time. That is the whole method, and it does not change no matter how cluttered the equation looks.

Inequalities work identically with one exception that the SAT tests constantly: multiplying or dividing by a negative number flips the inequality sign. Adding or subtracting a negative does not.

The more interesting questions ask not for a solution but for how many solutions exist. A linear equation has one solution when the variable survives, no solution when it cancels and leaves something false, and infinitely many when it cancels and leaves something true.

Seen as graphs, that is just two lines: crossing once, parallel and never meeting, or lying exactly on top of each other. The algebra and the picture are the same fact.

So when a question gives $ax + b = cx + d$ and asks for the value of $a$ making it have no solution, it is asking you to make the slopes equal while keeping the intercepts different.

Solve $4x + 7 = 2x - 5$.
Collect the variable on one side and the numbers on the other The variable survived, so there is exactly one solution.

Solve $-3x + 2 \leq 11$.
Subtract $2$ from both sides to get $-3x \leq 9$. Now divide by $-3$, which flips the sign Forgetting to flip gives $x \leq -3$, which is exactly the wrong half of the number line — and is always among the answer choices.