Fractional equations and inequality systems in Year 2

End of September, and in Year 2 of the art high school (liceo artistico) the review of equations and factorising are filling these weeks, up to October. The new block — fractional equations, linear inequalities and systems of inequalities — starts in November, and the test comes nine to twelve weeks from now, so between late November and mid-December. With 3 hours of maths a week, plus the 12 weekly hours of assignments in the specialist art subjects that give no respite at the art high school, there is no room to chase it afterwards: the moment to arrive with a method is now, before the chapter starts in class.

This post starts where factorising ends. If in class you are not yet clear on how to factorise a polynomial or simplify an algebraic fraction, the method — the decision tree, the special trinomial, the worked exercises — is in the post on factorising and algebraic fractions in Year 2: here I take factorising as done and start from fractional equations.

Fractional equation: the procedure in four steps

A fractional equation is an equation in which the unknown appears in a denominator. The procedure is always the same, in this order, with no shortcuts:

1. Condition of existence (C.E.). Every denominator must be different from zero. Factorise the denominators if needed, and write the condition on each factor.
2. Lowest common multiple (LCM) of the denominators. Bring all the terms, on the left and right of the equals sign, to the same denominator.
3. Whole equation. Once all the terms have the same denominator, you can compare just the numerators: this is where the fractional equation becomes a “normal” equation to solve with the rules you already know.
4. Check the C.E. against the solution. The step almost everyone skips. Every solution found at step 3 must be compared with the condition from step 1: if it coincides with an excluded value, it must be discarded, even if the algebraic calculation is correct.

Point 4 is the one flagged as critical for this syllabus, and not by chance: you can solve a fractional equation perfectly and still hand in a wrong solution, simply because you did not check that the value found does not make a denominator vanish.

Exercise 1 — fractional equation with a solution to discard

Solve:

Condition of existence. The denominator is $x-2$ on both sides:

Solution. The two sides have the same denominator, so for $x\ne 2$ I can compare the numerators directly:

I factorise: I look for two numbers that multiply to $10$ and add up to $-7$: they are $-2$ and $-5$.

Check against the C.E. The condition was $x\ne 2$. The solution $x=2$ coincides exactly with the excluded value: it must be discarded. Only this remains:

Direct check: on the left $\frac{5^2}{5-2}=\frac{25}{3}$; on the right $\frac{7\cdot 5-10}{5-2}=\frac{25}{3}$. They match, the solution holds.

If you want to practise this kind of check before moving on to inequalities, you will find the practice test here: maths test — first-degree inequalities. It is catalogued under Year 1 because in high schools with 5 hours of maths linear inequalities are already done in Year 1; at the art high school, with 3 hours, they come in Year 2, but the content is the same: use it without thinking you have landed on the wrong quiz.

Linear inequalities: what changes compared with equations

A linear inequality is solved with the same rules as an equation — adding or subtracting the same term on both sides, multiplying or dividing by the same number — with one difference, but a decisive one:

If you multiply or divide both sides by a negative number, the direction of the inequality flips.

A minimal example is enough to fix it:

I divide by $-2$ (negative): the $\ge$ becomes $\le$.

Not $x\ge -3$. If in an exercise you divide by a negative number and leave the direction as it was, the rest of the calculation can be perfect and the result is still wrong: it is the most frequent mistake on this topic, more than rational inequalities or systems.

How to write the solution set

An inequality, unlike an equation, usually has not a single solution but an interval of values. There are three equivalent ways to write it, and in a test you will be asked for the one in notation:

• In words: “all the values of $x$ greater than $-3$”.
• On the number line: an open dot on $-3$ (because $-3$ is not included) with an arrow to the right, to show that all larger values are fine.
• In interval notation: $x>-3$, or $(-3,+\infty )$.

The dot is filled when the value is included (inequality with $\ge$ or $\le$), open when it is excluded (inequality with $>$ or $<$). It is a small detail that in marked tests is worth marks of its own.

Systems of inequalities: the row diagram

A system of inequalities is solved like this: solve each inequality on its own, as seen above, and then intersect the intervals you found — that is, look for the values of $x$ that satisfy all the inequalities together, not just one.

The safest way not to get the intersection wrong is the row diagram: you draw a horizontal line for each inequality in the system, one below the other, with the same scale of values vertically; on each line you mark the part of the line that satisfies that inequality (dashed or shaded, depending on how your book draws it); at the bottom you add a summary row and take only the zone where all the rows are “full” at the same time. It is there, in that overlap, that most mistakes happen: people look at one row at a time and lose sight of which stretch is common to all.

Exercise 2 — system of three linear inequalities

Solve the system:

First inequality.

I divide by $2$, positive: the direction stays as it was.

Second inequality.

I divide by $-3$, negative: the direction flips, from $\ge$ to $\le$.

Third inequality.

I divide by $-1$, negative: the direction flips, from $<$ to $>$.

Row diagram. The three solutions are $x>2$, $x\le 3$, $x>-3$. On the number line:

• row 1: open dot on $2$, stretch to the right;
• row 2: filled dot on $3$, stretch to the left;
• row 3: open dot on $-3$, stretch to the right.

The condition $x>-3$ is already included in $x>2$ (every value greater than $2$ is also greater than $-3$), so it adds no restrictions. The stretch common to all three rows runs from just after $2$ (excluded) to $3$ (included):

In interval notation: $(2,3]$.

The impossible case and the always-true case

Two situations that the book asks you to recognise, without needing a long working:

• Impossible system. If one row asks for $x>3$ and another asks for $x<1$, there is no value that satisfies both: the intervals do not overlap anywhere. The solution is the empty set, $\varnothing$.
• Always-true inequality. If after simplifying you are left with an inequality that is true whatever $x$ is — for example $2x+1>2x-3$, which reduces to $1>-3$ — then it is true for every real $x$, and in a system that row adds no restriction.

Both cases are recognised by looking at the row diagram with the same attention you give to a normal intersection: they are not exceptions to fear, just a different result.

Before November

The weak point of this block is not the calculation — the rules are few and you have already seen them in equations — but the discipline of applying them in the right order: C.E. first of all in fractional equations, the direction flipped when needed in inequalities, the intersection read row by row in systems. If you also need to review the first year of this track, you will find the post on numbers and fractions in Year 1 at the art high school; the full formula sheet for the Year 2 syllabus is in the post on Year 2 maths.

To train on inequalities and systems you will find the same Year 1 practice test: maths test — first-degree inequalities — the correct answer is free, the guided solution is unlocked with your first lesson.

You will find all the subjects we cover on /materie/matematica. If you want a hand before the test, the free 30-minute introductory call is a good place to start → contact page.

For the framework of the whole year, with the chapters that come before this one, there is Year 2: the whole maths syllabus in a formula sheet; on the same theme, with the applied slant of the vocational track, I have written equations, inequalities and systems in Year 2 at the vocational school. When inequalities become fractional and the sign is studied factor by factor, the guide is rational inequalities in Year 3.