No. 13 Inequalities

A linear or quadratic inequality. The answer is an interval.

How to solve it

Linear inequalities

A linear inequality is solved almost like a linear equation: $3x - 5 > x + 1$. There is one difference, and it matters: when you multiply or divide by a negative number, the inequality sign flips.

Step by step

Move the $x$ terms to the left and the numbers to the right, changing the signs of the moved terms as in an equation.
Collect like terms to get $ax > b$ (or another inequality sign).
Divide both sides by $a$. If $a > 0$, the sign stays. If $a < 0$, it flips.
Write the answer as an interval: $x > 3$ is $(3, +\infty)$, and $x \le 3$ is $(-\infty, 3]$.

The number in front of $x$. Its sign decides whether the inequality flips. Why: multiplying $2 < 3$ by $-1$ gives $-2 > -3$.

Example: $-2x > 6$. Divide by $-2$ and flip the sign: $x < -3$. Answer: $(-\infty, -3)$.

Common mistakes

Dividing by a negative number without flipping the sign.
Mixing up brackets: with a strict sign ($>$ or $<$) the end is not included and the bracket is round; with $\ge$ or $\le$ it is included and the bracket is square.
Putting a square bracket next to infinity. Infinity always gets a round bracket.

Example

Quadratic inequalities

A quadratic inequality has a quadratic on the left: $x^2 - 5x + 6 \le 0$. Its graph is a parabola, and the answer is easiest to read straight off the picture: where the parabola is above the axis and where it is below.

Step by step

Move everything to the left so that the right side is zero.
Find the roots of the quadratic by solving $ax^2 + bx + c = 0$. At the roots the parabola crosses the axis.
Sketch the parabola: it opens upwards when $a > 0$ and downwards when $a < 0$.
For $>$ or $\ge$ take the parts where the parabola is above the axis; for $<$ or $\le$, where it is below.
Include the roots only when the inequality is not strict.

The leading coefficient. When it is positive, the parabola opens upwards: it is below the axis between the roots and above it outside. For $a < 0$ it is the other way round.

The roots of the quadratic, $x_1 < x_2$.

Example: $x^2 - 5x + 6 \le 0$. The roots are $2$ and $3$, the parabola opens upwards, and we need the points below the axis together with the roots: $[2, 3]$.

Common mistakes

Forgetting that a negative $a$ turns the parabola upside down, and taking the wrong part.
Mixing up «between the roots» and «outside». A quick check helps: substitute one number from your answer.
Not knowing what to do when $D < 0$. Then the quadratic has the same sign everywhere, and the answer is either every number or the empty set.