A linear equation and a linear inequality are solved with almost identical steps, and the exam expects you to know exactly where the one difference matters.

The equation

The book defines the form directly: "A linear equation looks like any equation that can be rearranged into the form: ax + b = 0 where x represents the variable, and a and b are constants." The method for solving one is to isolate x by undoing operations in reverse order. Take 6x - 9 = 51. Undo the subtraction first, by adding 9 to both sides: 6x = 60. Undo the multiplication last, by dividing both sides by 6: x = 10. Two steps, done in the reverse order of how the expression was built.

The inequality, and the one rule that changes

An inequality such as -4x + 7 <= 23 is solved with the same two steps, in the same order: subtract 7 from both sides to get -4x <= 16, then divide both sides by -4 to isolate x. Here is the one difference the exam is really testing: whenever you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol. Dividing -4x <= 16 by -4 gives x >= -4, not x <= -4. The number flips because a negative multiplier reverses the relative order of every quantity it touches; if 2 < 3, then -2 > -3, and the inequality symbol has to reflect that reversal or the solution set will be wrong.

You can check a solved inequality by testing a value on the boundary and one clearly outside the claimed solution set. At x = -4, the boundary, -4(-4) + 7 = 23, which satisfies "<= 23" exactly, confirming -4 belongs to the solution set. At x = -5, outside x >= -4, -4(-5) + 7 = 27, and 27 <= 23 is false, correctly excluding values below -4. This two-point check, one at the boundary and one outside it, is worth the fifteen extra seconds it costs, because sign errors on inequalities are one of the most common ways to lose an easy point on this exam.

It helps to see why the reversal rule is true, rather than just memorising it. Start from a true inequality, 2 < 3. Multiply both sides by -1: the left side becomes -2, the right side becomes -3. But -2 is greater than -3, not less than it, so the statement '-2 < -3' would be false. The only way to keep a true statement true after multiplying by a negative is to flip the direction of the symbol, giving -2 > -3, which is correct. The same logic applies whether you multiply by -1 or by any other negative number, and whether the inequality uses < or <=.

This is worth contrasting with solving an equation, where no such flip ever happens. An equation states that two quantities are equal, and multiplying or dividing both sides by the same number, positive or negative, preserves that equality without needing any adjustment. The flip rule belongs only to inequalities, precisely because an inequality is a statement about order, and multiplying by a negative number reverses order.

This reviewer adds the sign-flip check explicitly; the book states the form of a linear equation but does not walk through an inequality example, so treat the reversal rule above as this reviewer's addition, matching the standard algebra rule taught in every high school text.