Show the work to solve |2x+5| -1 < 6 and describe the graph of the solution in a complete sentence.
| 2x + 5 | - 1 < 6
| 2x + 5 | < 6 + 1
| 2x + 5 | < 7
Now, you should split the equation into two (due to the absolute value).
Remember that |x| is the equivilant of x or -x
So solve for:
(2x + 5) < 7
and
-(2x + 5) < 7
The first equation becomes:
2x < 7 - 5
2x < 2 Divide both sides with 2
x < 1
The second equation becomes:
-(2x + 5) < 7
-2x - 5 < 7
-2x < 7 + 5
-2x < 12 Divide both sides with -2
x > -6
Becouse if you multiply or divide both sides of an inequality by a negative number, you MUST reverse the direction of the inequality sign!
Solution:
-6 < x < 1
If you want to see graph of | 2x + 5 | < 7
in google type:
functions graphs online
When you see list of results click on:
rechneronline.de/function-graphs/
When page be open in blue rectangle type:
abs(2x+5)
in gray rectangle type 7
Set:
Range x-axis from -8 to 2
Range y-axis from -2 to 8
and click option Draw
Or go on:
wolfram alpha dot com
then type |2x+5| < 7 and click option =
Also you can construct graph:
For x = -6
| 2x + 5 | = | 2 * ( -6 ) + 5 | = | -12 + 5 | = | -7 | = 7
For x = -5
| 2x + 5 | = | 2 * ( -5 ) + 5 | = | -10 + 5 | = | -5 | = 5
For x = 0
| 2x + 5 | = | 2 * 0 + 5 | = | 0 + 5 | = | 5 | = 5
For x = 1
| 2x + 5 | = | 2 * 1 + 5 | = | 2 + 5 | = | 7 | = 7
etc.
To solve the inequality |2x+5| -1 < 6, we need to isolate the absolute value expression in the inequality and consider the two separate cases when the expression inside the absolute value is positive and negative.
1. When 2x+5 is positive:
|2x+5| -1 < 6
2x+5 - 1 < 6
2x + 4 < 6
2x < 2
x < 1
2. When 2x+5 is negative:
|-(2x+5)| -1 < 6
|-2x-5| -1 < 6
-2x-5 - 1 < 6
-2x - 6 < 6
-2x < 12
x > -6
The solution to the inequality is -6 < x < 1, which can be written as (-6, 1).
The graph of the solution is a number line that is shaded between -6 and 1, indicating that any value of x within this interval satisfies the inequality |2x+5| -1 < 6.