The variable on both sides

Some­times the vari­able appears on both sides of the equals sign, as in $\displaystyle \frac{x}{2} + 3 = \frac{x}{3} + 5$. Before you can find $x$, you must bring all the $x$ terms together on one side and all the plain num­bers on the other. In this les­son you will do exactly that, with frac­tions, dec­i­mals and brack­ets.

Gather the terms#

A term is one part of a side, joined to the rest by $+$ or $-$. In $\displaystyle \frac{x}{2} + 3$ there are two terms: $\displaystyle \frac{x}{2}$ and $3$.

1. Move every term with $x$ to the left side. Change its sign as it crosses.
2. Move every plain num­ber to the right side. Change its sign as it crosses.
3. Join the like terms on each side.
4. Fin­ish as a one-step equa­tion.

Like terms are joined just as in alge­bra: $3a + 2a = 5a$ and $10a - 8a = 2a$.

Exam­ple 1#

Solve $\displaystyle \frac{x}{2} + 3 = \frac{x}{3} + 5$.

The $\displaystyle \frac{x}{3}$ moves left and becomes $\displaystyle -\frac{x}{3}$. The $3$ moves right and becomes $-3$.

$$\displaystyle \begin{aligned} \frac{x}{2} + 3 &= \frac{x}{3} + 5 \\ \Rightarrow \frac{x}{2} - \frac{x}{3} &= 5 - 3 \\ \Rightarrow \frac{3x - 2x}{6} &= 2 \\ \Rightarrow \frac{x}{6} &= 2 \\ \Rightarrow x &= 12 \end{aligned}$$

Check: LHS $\displaystyle = \frac{12}{2} + 3 = 6 + 3 = 9$. RHS $\displaystyle = \frac{12}{3} + 5 = 4 + 5 = 9$. LHS $=$ RHS.

Exam­ple 2#

Solve $1.5x + 2 = 0.5x + 7$.

$$\begin{aligned} 1.5x + 2 &= 0.5x + 7 \\ \Rightarrow 1.5x - 0.5x &= 7 - 2 \\ \Rightarrow x &= 5 \end{aligned}$$

Here $1.5x - 0.5x = 1x$, which we write sim­ply as $x$.

Exam­ple 3#

Solve $\displaystyle \frac{3x}{4} - 1 = \frac{x}{2} + \frac{1}{4}$.

$$\displaystyle \begin{aligned} \frac{3x}{4} - 1 &= \frac{x}{2} + \frac{1}{4} \\ \Rightarrow \frac{3x}{4} - \frac{x}{2} &= \frac{1}{4} + 1 \\ \Rightarrow \frac{3x}{4} - \frac{2x}{4} &= \frac{5}{4} \\ \Rightarrow \frac{x}{4} &= \frac{5}{4} \\ \Rightarrow x &= 5 \end{aligned}$$

Exam­ple 4#

Solve $\displaystyle 5x - \frac{2}{3} = 2x + \frac{1}{3}$.

$$\displaystyle \begin{aligned} 5x - \frac{2}{3} &= 2x + \frac{1}{3} \\ \Rightarrow 5x - 2x &= \frac{1}{3} + \frac{2}{3} \\ \Rightarrow 3x &= 1 \\ \Rightarrow x &= \frac{1}{3} \end{aligned}$$

Equa­tions with brack­ets#

Open the brack­ets first. A num­ber out­side a bracket mul­ti­plies every term inside it.

Exam­ple 5#

Solve $\displaystyle 2\left(x - \frac{1}{2}\right) = 3(x + 1)$.

$$\displaystyle \begin{aligned} 2\left(x - \frac{1}{2}\right) &= 3(x + 1) \\ \Rightarrow 2x - 1 &= 3x + 3 \\ \Rightarrow 2x - 3x &= 3 + 1 \\ \Rightarrow -x &= 4 \\ \Rightarrow x &= -4 \end{aligned}$$

If $-x = 4$, then $x = -4$. Change the sign on both sides.

Check: LHS $= 2(-4 - 0.5) = 2 \times (-4.5) = -9$. RHS $= 3(-4 + 1) = 3 \times (-3) = -9$. It works.

Exam­ple 6#

Solve $0.2(x + 4) = 0.5x - 0.7$.

$$\displaystyle \begin{aligned} 0.2(x + 4) &= 0.5x - 0.7 \\ \Rightarrow 0.2x + 0.8 &= 0.5x - 0.7 \\ \Rightarrow 0.2x - 0.5x &= -0.7 - 0.8 \\ \Rightarrow -0.3x &= -1.5 \\ \Rightarrow x &= \frac{-1.5}{-0.3} = \frac{15}{3} = 5 \end{aligned}$$

A minus divided by a minus gives a plus.

Exam­ple 7: a minus sign before a bracket#

Solve $5(x + 0.4) - 2(x - 1) = 13$.

The $-2$ mul­ti­plies both terms in its bracket: $-2 \times x = -2x$ and $-2 \times (-1) = +2$.

$$\begin{aligned} 5(x + 0.4) - 2(x - 1) &= 13 \\ \Rightarrow 5x + 2 - 2x + 2 &= 13 \\ \Rightarrow 3x + 4 &= 13 \\ \Rightarrow 3x &= 9 \\ \Rightarrow x &= 3 \end{aligned}$$

Remem­ber. Open brack­ets first. Then $x$ terms to the left, num­bers to the right, chang­ing the sign of any­thing that crosses.

Com­mon mis­takes#

• Mov­ing a term with­out chang­ing its sign.
• Mul­ti­ply­ing only the first term inside a bracket. $0.2(x + 4)$ is $0.2x + 0.8$.
• For­get­ting that a minus before a bracket changes every sign inside: $-2(x - 1) = -2x + 2$.
• Stop­ping at $-x = 4$. The answer is $x = -4$.

Prac­tice#

1. Solve $\displaystyle 3x + \frac{1}{2} = x + \frac{5}{2}$.
2. Solve $2.5x - 3 = x + 1.5$.
3. Solve $\displaystyle \frac{x}{3} + 2 = \frac{x}{4} + 3$.
4. Solve $4x - 1.6 = 6x + 0.4$.
5. Solve $\displaystyle \frac{2x}{5} - 1 = \frac{x}{10} + 2$.
6. Solve $\displaystyle 7x - \frac{3}{4} = 3x + \frac{5}{4}$.
7. Solve $\displaystyle x - \frac{2}{3} = \frac{1}{6} - x$.
8. Solve $\displaystyle 3(x - 2) = 2\left(x + \frac{1}{2}\right)$.
9. Solve $0.5(2x - 6) = 0.3x + 4$.
10. Solve $4(x - 0.5) - 3(x - 2) = 7$.

Answers#