Lessons · Teaching · linear equations, inequalities and what a function is

Solving, rearranging, and what a function is

You need to solve a linear equation, solve and graph a simple inequality, work with a formula by rearranging it, read a straight-line graph for its slope and intercept, and recognise a function as a rule that gives exactly one output for each input.

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What it is for

This is the section that stops people, and it stops them because it was last met at sixteen and never used since. It is also perfectly finite: there are about six things here and none of them is deep. The gap is almost always confidence and practice rather than capacity.

How to think about it

Solve by undoing, in the reverse of the order the expression was built. Keep both sides equal by doing the same thing to each. For an inequality, everything is the same except that multiplying or dividing by a negative number flips the sign, which is the one rule worth memorising outright.

Worked example

3x + 7 = 22, so 3x = 15 and x = 5

Undo the plus 7, then undo the times 3. The reverse of how it was built.

Slope is rise over run: from (1, 2) to (4, 11), the slope is 9 / 3 = 3

Change in y over change in x, and 11 - 2 = 9 while 4 - 1 = 3.

-2x > 6 gives x < -3, because dividing by a negative flips the sign

The single rule in inequalities that is not the same as in equations.

A function gives one output per input, so a vertical line crosses its graph at most once

Which is the whole of the vertical line test and the whole of the definition.

Your turn

Name what happens to an inequality sign when you divide by a negative.

Dividing by a negative number  the inequality sign

The trap

Doing something to one side only. Every legal move in an equation is done to both sides, and the errors that survive a check are nearly all a step applied once.

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