Inequalities

Starting point: what you already know

• Solve a linear inequality like an equation, but flip the sign if you multiply or divide by a negative number: $-2x<6$ gives $x>-3$.
• Quadratic inequalities (Mathematics Advanced): to solve $x^2-x-6<0$, factorise to $(x-3)(x+2)<0$, sketch the smile parabola and read where it is below the axis: $-2<x<3$.
• $|x|$ is the distance of $x$ from 0. So $|x-a|$ is the distance between $x$ and $a$.

Cubic inequalities

A cubic like $(x+1)(x-2)(x-4)$ has three roots. A positive cubic (leading coefficient positive) starts low on the left and ends high on the right, crossing the axis at each single root.

Inequalities with x in the denominator

You cannot just multiply both sides by $x$, because you do not know if $x$ is positive or negative. Instead, multiply both sides by the square of the denominator, which is always positive.

Absolute value inequalities

Read $|x-a|$ as the distance from $x$ to $a$. Then:

• $|x-a|<k$ means $x$ is within $k$ of $a$: $a-k<x<a+k$.
• $|x-a|>k$ means $x$ is further than $k$ from $a$: $x<a-k$ or $x>a+k$.

Quick summary

• Find where the expression equals zero, sketch, then read above the axis for $>0$ or below it for $<0$.
• With $x$ in the denominator, multiply both sides by the denominator squared and leave out the value that makes it zero.
• $|x-a|<k$ means $a-k<x<a+k$. $|x-a|>k$ means $x<a-k$ or $x>a+k$.

Quiz