Solving Sentences Involving "Plus or Minus"

The sentence ‘$\,x = \pm 3\,$’ is a convenient shorthand for ‘$\,x = 3\ \text{ or }\ x = -3\,$’ .
Sentences like this are important when solving absolute value equations.
The sentence ‘$\,x = \pm 3\,$’ is read aloud as ‘$\,x\,$ is plus or minus three’ or ‘$\,x\,$ equals plus or minus three’ .
This web exercise gives you practice working with ‘plus or minus’ sentences.

When working with sentences involving plus or minus ($\,\pm\,$), you have two choices:

break into an ‘or’ sentence immediately
wait until the last step to break into an ‘or’ sentence

The examples below illustrate both approaches.

EXAMPLE: Break into an ‘or’ sentence immediately

Solve:

$2x - 1 = \pm 5$

Solution:
Be sure to write a nice, clean list of equivalent sentences.

$2x - 1 = \pm 5$	(original sentence)
$2x - 1 = 5\ \text{ or }\ 2x - 1 = -5$	(expand the shorthand notation)
$2x = 6\ \text{ or }\ 2x = -4$	(add $\,1\,$ to both sides of both equations)
$x = 3\ \text{ or }\ x = -2$	(divide both sides of both equations by $\,2\,$)

EXAMPLE: Wait until the last step to break into an ‘or’ sentence

Solve:

$2x - 1 = \pm 5$

Solution:

$2x - 1 = \pm 5$	(original sentence)
$2x = \pm 5 + 1$	(add $\,1\,$ to both sides—you cannot simplify anything on the right!)
$\displaystyle x = \frac{\pm 5 + 1}{2}$	(divide both sides by $\,2\,$)
$\displaystyle x = \frac{5 + 1}{2}\ \text{ or }\ x = \frac{-5 + 1}{2}$	(expand the shorthand; you can probably skip this step and jump right to the next one)
$\displaystyle x = 3\ \text{ or }\ x = -2$	(simplify)

The method you choose to use is entirely up to you!

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Solving Absolute Value Equations

Solve the given absolute value sentence.
Write the result in the most conventional way.

For more advanced students, a graph is displayed.
For example, the sentence $\,2x - 1 = \pm 5\,$
is optionally accompanied by the graph of $\,y = 2x - 1\,$ (the left side of the equation, dashed green)
and the graph of $\,y = \pm 5\,$ (the right side of the equation, solid purple).
In this example, you are finding the values of $\,x\,$ where the green graph intersects the purple graph.
Click the “show/hide graph” button if you prefer not to see the graph.