Equations, inequalities and systems

Linear equations

A linear equation is one that can be written in the form $ax+b=0$ with $a\ne 0$. To solve it you must get $x$ on its own on one side, and the golden rule is that whatever you do to one side you do to the other: add or subtract the same quantity, or multiply or divide by the same non-zero number. Each operation produces an equivalent equation, with the same solution.

For example, $4(x-2)=2x+6$ becomes $4x-8=2x+6$, so $2x=14$ and $x=7$. Check: $4\cdot 5=20$ and $2\cdot 7+6=20$. A linear equation normally has a single solution. If simplifying leaves something false, such as $0=3$, it has no solution; if it leaves something always true, such as $0=0$, it has infinitely many.

Linear inequalities

An inequality uses $<$, $\le$, $>$ or $\ge$ instead of the equals sign. It is solved with the same steps as an equation, with one exception:

The reason is simple: $2<5$ is true, but multiplying by $-1$ gives $-2$ and $-5$, and now $-2$ is the larger, so you must write $-2>-5$.

The solution of an inequality is a set of infinitely many values, and it is written using intervals. In IB notation, a bracket that faces outwards, as in $]3,\infty [$, leaves the endpoint outside the interval, and a bracket that faces inwards, as in $[-1,4]$, includes it. With a strict inequality ($<$, $>$) the endpoint is not included; with $\le$ or $\ge$ it is. Infinity never takes a closed bracket. On the number line, an included endpoint is marked with a filled dot and an excluded one with an open dot.

For example, $9-2x\ge 3$ gives $-2x\ge -6$ and, dividing by $-2$, $x\le 3$: the solution set is $]-\infty ,3]$.

Systems of two linear equations

A system of two equations with two unknowns asks for a pair $(x,y)$ that satisfies both equations at the same time. Geometrically, each equation is a line, and the solution is their point of intersection. There are three possibilities:

• The lines intersect at one point: unique solution.
• The lines are parallel: no solution, as in $x+y=3$ together with $2x+2y=8$.
• The two equations describe the same line: infinitely many solutions, as in $x+y=3$ together with $2x+2y=6$.

Almost all exam systems have a unique solution. By hand there are two methods.

With $x+2y=7$ and $3x-y=7$: from the first, $x=7-2y$. In the second, $3(7-2y)-y=7$, that is, $21-7y=7$, so $y=2$ and $x=7-4=3$. The solution is $(3,2)$.

With $2x+3y=12$ and $5x-2y=11$ we eliminate $y$: multiply the first by $2$ and the second by $3$, giving $4x+6y=24$ and $15x-6y=33$. Adding, $19x=57$, so $x=3$, and substituting into the first, $6+3y=12$, so $y=2$. Again $(3,2)$.

The quadratic equation

A quadratic equation has the form $a{x}^2+bx+c=0$ with $a\ne 0$, and its solutions are called roots. Before solving it, move everything to one side and leave $0$ on the other.

If the equation is incomplete, it can be solved without the formula:

• If the $x$ term is missing ($a{x}^2+c=0$): solve for $x^2=-\frac{c}{a}$. There are two opposite roots if $-\frac{c}{a}>0$ and no real roots if it is negative.
• If the constant term is missing ($a{x}^2+bx=0$): take out a common factor, $x(ax+b)=0$. A product is zero only if one of its factors is zero, so $x=0$ or $x=-\frac{b}{a}$.

If the equation is complete, you can factorise (looking for two numbers with product $c$ and sum $b$ when $a=1$) or apply the formula, which always works.

For example, in $2x^2-7x+3=0$ we have $a=2$, $b=-7$, $c=3$, and $$x=\frac{7\pm \sqrt{49-24}}{4}=\frac{7\pm 5}{4}\Rightarrow x=3\text{ or }x=\frac{1}{2}$$

This has a graphical reading: the parabola $y=a{x}^2+bx+c$ meets the $x$-axis at two points, touches it at only one (its vertex) or does not reach it, depending on the sign of $\Delta$. For example, $x^2+2x+5=0$ has $\Delta =4-20=-16<0$, and therefore has no real solutions.