Algebra — Mathslive.ie

Section 1 of 3

Algebraic Fractions

To add or subtract algebraic fractions, put everything over a common denominator, then tidy the top.

Worked example — subtract two fractions

Simplify $\dfrac{2x+1}{3} - \dfrac{x-1}{5}$.

Common denominator $15$: $\dfrac{5(2x+1) - 3(x-1)}{15}$

$= \dfrac{10x+5 - 3x+3}{15}$

$\dfrac{7x+8}{15}$

Worked example — with a variable denominator

Simplify $\dfrac{5}{2x-1} - \dfrac{3}{7}$.

Common denominator $7(2x-1)$: $\dfrac{5(7) - 3(2x-1)}{7(2x-1)}$

$= \dfrac{35 - 6x + 3}{14x-7}$

$\dfrac{38-6x}{14x-7}$

Worked example — evaluate a fraction expression

Evaluate $\dfrac{2x+1}{4} - \dfrac{3x-4}{3}$ when $x = \tfrac12$; give the answer as $\tfrac{a}{b}$.

$\dfrac{2(\frac12)+1}{4} - \dfrac{3(\frac12)-4}{3} = \dfrac{2}{4} - \dfrac{-\frac52}{3}$

$= \dfrac12 + \dfrac{5}{6} = \dfrac{3}{6} + \dfrac{5}{6}$

$\dfrac{8}{6} = \dfrac{4}{3}$

Section 2 of 3

Change of Subject

To make a new letter the subject, undo the operations — multiply out, gather that letter on one side, then divide.

Worked example — a simple rearrange

Given $p = \dfrac{x+2y}{3}$, express $y$ in terms of $x$ and $p$.

Multiply by $3$: $3p = x + 2y$

$2y = 3p - x$

$y = \dfrac{3p-x}{2}$

Now you try

You try

Given $2(2q - 7p) = q(3p - q)$, express $p$ in terms of $q$.

Pen and paper out — try it before you reveal.

Expand: $4q - 14p = 3pq - q^2$

Gather $p$: $-14p - 3pq = -q^2 - 4q \Rightarrow 14p + 3pq = q^2 + 4q$

Factor: $p(14 + 3q) = q^2 + 4q$

$p = \dfrac{q^2 + 4q}{14 + 3q}$

Section 3 of 3

Inequalities

Solve like an equation — but flip the sign whenever you multiply or divide by a negative.

You try

List the solution set of $-3x - 3 > x - 12$, $x \in \mathbb{N}$.

Pen and paper out — try it before you reveal.

$-3x - x > -12 + 3 \Rightarrow -4x > -9$

Divide by $-4$ (flip): $x < \dfrac{9}{4} = 2.25$

Natural numbers less than $2.25$: $1, 2$

$\{1,\ 2\}$

You try

Graph the solution of $-9 \le 2x - 5 < 7$, $x \in \mathbb{Z}$.

Pen and paper out — try it before you reveal.

Left: $-9 \le 2x - 5 \Rightarrow -4 \le 2x \Rightarrow x \ge -2$

Right: $2x - 5 < 7 \Rightarrow 2x < 12 \Rightarrow x < 6$

$-2 \le x < 6$, i.e. $\{-2,-1,0,1,2,3,4,5\}$

Expanding Brackets

Multiply out brackets by multiplying every term in the first by every term in the second, then collect like terms.

Worked example — two brackets

Expand $(3x+1)(2x+9)$.

$3x(2x+9) + 1(2x+9)$

$6x^2 + 27x + 2x + 9$

$6x^2 + 29x + 9$

Worked example — a perfect square

Expand $(6a-1)^2$.

$(6a-1)(6a-1) = 36a^2 - 6a - 6a + 1$

$36a^2 - 12a + 1$

You try

Expand $(5p-3)(6p-7)$.

Every term times every term, then collect.

$30p^2 - 35p - 18p + 21$

$30p^2 - 53p + 21$

You try

Expand $(3q-5)(q^2-6q-1)$.

Multiply the $3q$ across, then the $-5$, then collect like terms.

$3q(q^2-6q-1) - 5(q^2-6q-1)$

$3q^3 - 18q^2 - 3q - 5q^2 + 30q + 5$

$3q^3 - 23q^2 + 27q + 5$

That’s Algebra.

Algebraic fractions, changing the subject, and inequalities — the core algebra skills for Junior Cert.