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
| $2x - 1 = \pm 5$ | 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
| $2x - 1 = \pm 5$ | 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.