Linear equations and inequalities in one variable can always be solved using this method:

- Simplify both sides (if necessary).
- If there is a term on the right side with the variable in it, then subtract this term from both sides (and simplify).
- If there is a constant term on the left side, then subtract this term from both sides (and simplify).
- If there is now a negative coefficient on the variable on the left side, then take the opposite of both sides (and simplify).
- If there is now a positive coefficient on the variable on the left side, then divide both sides by that coefficient (and simplify).

- 5
*x*+ 2 = 3*x*+ 6 — original equation; - 2
*x*+ 2 = 6 — subtract 3*x*from both sides; - 2
*x*= 4 — subtract 2 from both sides; *x*= 2 — divide both sides by 2.

The next example needs to be simplified first:

- 3(2
*a*+ 4) = 2*a*+ 2(*a*+ 9) — original equation; - 6
*a*+ 12 = 4*a*+ 18 — simplify both sides; - 2
*a*+ 12 = 18 — subtract 2*a*from both sides; - 2
*a*+ 12 = 18 — subtract 2*a*from both sides; - 2
*a*= 6 — subtract 12 from both sides; *a*= 3 — divide both sides by 2.

Here's an example with some negative numbers:

- 2
*t*− 8 = 5*t*− 5 — original equation; - −3
*t*− 8 = −5 — subtract 5*t*from both sides; - −3
*t*= 3 — add 8 (same as subtracting −8) from both sides; - 3
*t*= −3 — take the opposite of both sides; *t*= −1 — divide both sides by 3.

Here's an example where you can skip one step:

- 4
*x*= 2*x*+ 6 — original equation; - 2
*x*= 6 — subtract 2*x*from both sides; *x*= 3 — divide both sides by 2.

Here's a linear inequality; the basic technique is the same:

- 5
*y*+ 4 < 2*y*− 2 — original inequality; - 3
*y*+ 4 < −2 — subtract 2*y*from both sides; - 3
*y*< −6 — subtract 4 from both sides; *y*< −2 — divide both sides by 3.

Here's an example where I have to change the direction of the inequality:

- 2
*c*+ 5 ≥ 3*c*+ 7 — original inequality; - −
*c*+ 5 ≥ 7 — subtract 3*c*from both sides; - −
*c*≥ 2 — subtract 5 from both sides; *c*≤ −2 — take the opposite of both sides.

Here is an example of a false statement:

- 2
*n*+ 4 = 2(*n*+ 4) — original expression; - 2
*n*+ 4 = 2*n*+ 8 — simplify both sides; - 4 = 8 — subtract 2
*n*from both sides; - False — 4 < 8 in fact.

Here is an example of a true statement:

- 3
*p*+ 6 = 3(*p*+ 2) — original expression; - 3
*p*+ 6 = 3*p*+ 6 — simplify both sides; - 6 = 6 — subtract 3
*p*from both sides; - True — 6 = 6 in fact.

Go back to the MATH-0950-ES32 homepage.

This web page was written in 2007 by Toby Bartels. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-0950/2007SP/linear/`

.