Answer:
The result can be shown in multiple forms.
Inequality Form:
−12< x < 7
Interval Notation:
(−12,7)
Step-by-step explanation:
first write |2x+5|<19 as a piece wise
2x+5≥0
To find the interval for the first piece, find where the inside of the absolute value is non-negative.
[tex]\frac{2x}{2}[/tex] ≥[tex]\frac{5}{2}[/tex] divide
x ≥−[tex]\frac{5}{2}[/tex] cancel common factor of 2
x ≥ −5/2.
To find the interval for the second piece, find where the inside of the absolute value is negative.
2x+5 < 0
substract by 5
2x < −5
divide by 2x
2x/2< −5/2
simplify to left side
x < −5 / 2
Simplify the right side.
x < −5/2
In the piece where 2x+5 is negative, remove the absolute value and multiply by −1
− (2x+5) < 19
Write as a piecewise.
{2x+5 <19 x ≥ − 5/2
− (2x+5) < 19 x <− 5/2
Simplify −(2x+5)<19.
{2x+5<19 x≥−5/2
−2x−5<19 x<−5/2
Solve 2x+5 < 19 when x ≥ −5/2.
2x+5<19 for x.
Move all terms not containing x to the right side of the inequality.
2x < 14
divide by 2
x < 7
Find the intersection of x < 7 and x ≥−5/2.
−5/2 ≤x <7
Solve −2x−5 <19 when x <−5/2.
Move all terms not containing x to the right side of the inequality.
−2x<24
divide by -2
x>−12
Find the intersection of x > −12 and x <−5/2.
Find the union of the solutions.
−12 < x < 7
