Hong Kong
Stage 4 - Stage 5

# Solve Quadratic Inequalities

Lesson

Let's say we have the inequality $x+4<2x-6$x+4<2x6.

By now we know that we can use our elementary operations $+$+,$-$,$\times$×,$\div$÷​ to solve inequalities like this in exactly the same way that we solve equations, as long as we remember to reverse the inequality sign whenever we multiply or divide by a negative number. Hence, we could get our answer like this:

 $x+4$x+4 $<$< $2x-6$2x−6 $x$x $<$< $2x-10$2x−10 $-x$−x $<$< $-10$−10 $x$x $>$> $10$10

But what question are we really being asked when we are asked to solve $x+4<2x-6$x+4<2x6? We are being asked for values of $x$x at which $x+4$x+4 is less than $2x-6$2x6.

We can answer this question graphically by considering two lines $y_1=x+4$y1=x+4 and $y_2=2x-6$y2=2x6

### Solving using Test Values

You may have noticed by now that if a parabola has two distinct roots (when its discriminant is positive), it will always be on one side of the $x$x-axis between the two intercepts, and on the other side of the $x$x-axis outside the two intercepts. Parabolas are either positive between the two intercepts and negative outside of them, or negative between the two intercepts and positive outside of them.