Simultaneous equations and inequalities

Solving two equations at once (including a line meeting a curve), and solving linear and quadratic inequalities — including the sign-flip and the inside/outside rule for quadratics.

Pure · Algebra & functions

Simultaneous equations and inequalities

Solving two equations at once (including a line meeting a curve), and solving linear and quadratic inequalities — including the sign-flip and the inside/outside rule for quadratics.

Why it works

Simultaneous equations ask for the values that satisfy both statements at once — graphically, where the two graphs cross.

Two linear equations. Eliminate one variable (add/subtract the equations so a

variable cancels) or substitute one into the other. One intersection point, so one

(x, y)

pair.

One linear, one quadratic. Rearrange the linear equation for one variable

and substitute it into the other. You'll land on a single quadratic; solve it, then substitute each

x

back to get its partner

y

. A line can cut a curve in up to two places, so expect up to two pairs — and always quote them as pairs, not a loose list of numbers.

Inequalities behave almost exactly like equations — you can add, subtract, multiply and divide — with one extra rule: *multiplying or dividing both sides by a negative reverses the sign*. Why? On the number line

3 < 5

, but multiplying by

-1

sends them to

-3

and

-5

, and now

-3 > -5

. The order flips. (Avoid the trap entirely by keeping the variable's coefficient positive.)

Quadratic inequalities can't be handled by "just rearranging" — you need the shape of the parabola. The method:

Solve the matching equation to find where the parabola crosses the axis (its

roots).

Sketch the parabola (a positive $x^2$ opens upward).
Read off where it is above or below zero.

For an upward parabola,

y

is negative between the roots and positive outside them. So if the roots are

-2

and

3

$x^2 - x - 6 < 0$ (below the axis) → $-2 < x < 3$ (between),
$x^2 - x - 6 > 0$ (above the axis) → $x < -2$ or $x > 3$ (outside).

The sketch is what stops you writing the impossible "

3 < x < -2