How do you solve and write the following in interval notation: #2x^2 - 7x - 4<=0#?

1 Answer
rzp7
Apr 1, 2017

The solution to this inequality is: #[-1/2,4]# (see explanation for an in-depth process of how to solve the inequality).

Explanation:

We start by assuming #<=# is an #=# and solving as such:
#2x^2-7x-4=0#
#(2x+1)(x-4)=0#
#x=-1/2, 4#

These points, #x=-1/2# and #x=4#, the solutions to the equation, give us the bounds of the intervals for the inequality.

When #x<-1/2#, the function #(f(x)=2x^2-7x-4)# is positive.
When #-1/2<##x<4#, the function is negative.
When #x>4#, the function is positive.

Since we're looking for where #f(x)<=0#, the interval of the solution is #-1/2<=##x<=4#, which in interval notation is: #[-1/2,4]#.

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