Brackets, fractions and special equations

A long equation – with brackets on both sides or with fractions – is simplified until it becomes a simple one, like $\displaystyle {3x}={12}$. Sometimes on the way $\displaystyle x$ disappears from both sides, and then the equation has no solution, or every $\displaystyle x$ is a solution of it.

A minus before brackets changes the sign of every term inside them: $\displaystyle {-(2x-5)}={-2x+5}$.
The example below opens the brackets on both sides, collects like terms, and only then solves:

Check: the first side is $\displaystyle {4\cdot1-3}={1}$, and the second $\displaystyle {3\cdot5-14}={1}$.

Solve.

An equation with fractions becomes one without fractions: multiply both sides by a common denominator of all the fractions.
In the example below the denominators are 2 and 5, so we multiply by 10:

• $\displaystyle {10\cdot\frac{x+1}{2}}={5(x+1)}$: $\displaystyle {10:2}={5}$, and the denominator is gone.
• The numerator $\displaystyle x-2$ goes into brackets: the minus before the fraction applies to the whole numerator.
• 3 is multiplied by 10 too: every term on both sides is multiplied.

Solve.
In the first step, multiply both sides by the common denominator.

When all the terms with $\displaystyle x$ are moved to one side, sometimes they cancel out and $\displaystyle 0x$ is left. $\displaystyle 0x$ is 0, whatever $\displaystyle x$ is:

• $\displaystyle {0x}={5}$: no number times 0 gives 5 – no solution.
• $\displaystyle {0x}={0}$: every number times 0 gives 0 – every $\displaystyle x$ is a solution.

The first equation has no solution. In the second every $\displaystyle x$ is a solution: the two sides are equal expressions.

An equation asks: for which $\displaystyle x$ are the two sides equal? “No solution” and “every $\displaystyle x$” answer that question in a story too.
Company A charges 30 ₪ a month, plus 2 ₪ for each gigabyte.
Company B charges 20 ₪ a month, plus 2 ₪ for each gigabyte.
For how many gigabytes is the price the same at both companies?

Let $\displaystyle x$ be the number of gigabytes: $\displaystyle {30+2x}={20+2x}$. Subtracting $\displaystyle 2x$ and 20 from both sides gives $\displaystyle {0x}={10}$ – no solution. Company A always costs 10 ₪ more, for any amount.

On the graph: $\displaystyle {y}={2x+30}$ and $\displaystyle {y}={2x+20}$ (dashed) are lines with the same slope, 2. Both rise by 2 for each step of $\displaystyle x$, so the gap between them stays 10 – they are parallel and never meet.

Had company B’s fixed price been 30 ₪ too, the equation would be $\displaystyle {30+2x}={30+2x}$ – every $\displaystyle x$ is a solution: the two graphs are the same line, and the price is equal for any amount.

Solve.
Each equation has one solution, no solution, or every $\displaystyle x$ is a solution of it.

A basketball club charges a 50 ₪ joining fee plus 40 ₪ a month.
A swimming club charges an 80 ₪ joining fee plus 40 ₪ a month.
After how many months will the two clubs have cost the same?
Write an equation and solve it.
What does the answer say about the two clubs?

Noa solved the equation $\displaystyle {2(x+3)}={2x+6}$.
She got $\displaystyle {2x+6}={2x+6}$, and then $\displaystyle {0x}={0}$.
She wrote: “So $\displaystyle {x}={0}$.”
What was her mistake?
What is the right answer?