Answer:
[tex]\displaystyle a > 0[/tex]
Step-by-step explanation:
We are given the function:
[tex]\displaystyle f(x) = 2|x+1| + a[/tex]
And we want to find all values of a such that the are no x-intercepts.
Recall that an x-intercept is when the graph crosses the x-axis. So, let y = 0:
[tex]\displaystyle \left(0\right) = 2|x+1| + a[/tex]
Isolate the absolute value:
[tex]\displaystyle - \frac{1}{2} a = |x+1|[/tex]
Absolute value only outputs zero or positive values. So, in order for the function to have no x-intercepts, the left-hand side must be negative. That is:
[tex]\displaystyle -\frac{1}{2} a < 0[/tex]
Solve for a:
[tex]\displaystyle a > 0[/tex]
In conclusion, the value of a for which the given function has no solution is all real numbers greater than zero.
