# Solving Sentences Involving ‘Plus or Minus’

The sentence  ‘$\,x = \pm 3\,$’  is a convenient shorthand for  ‘$\,x = 3\,$ 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!

## Concept Practice

Solve the given ‘plus or minus’ value sentence. Write the result in the most conventional way.

For more advanced students, a graph is available. 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 to toggle the graph.

Solve: