These questions are pertaining to College Algebra

Question

(3x - 4)(x + 2)(x + 6) ≤ 0

 

1. On a single number line, do the following:

 

a) Graph the intervals that satisfy the inequality, including endpoints. The is the graph of the solution set of the inequality.

b) Write the solution set in interval notation.

Answer 1

This is a positive cubic graph that will be on or below the x-axis.  So first, we need to find the zeros.

 

(3x - 4)(x + 2)(x + 6) = 0

 

 

Set these factors equal to zero.

 

3x - 4 = 0          and            x + 2 = 6              and           x + 6 = 0

 

 

Solve for x in each case.

 

x = 4/3           ,                  x = -2            ,         and         x = -6

 

 

Now lets put these zeros in order from least to greatest.  These zeros are our reference points.

 

x = -6                 x = -2                x = 4/3

 

 

In a positive cubic graph, the function values decrease as x approaches negative infinity, and increase as x approaches positive infinity.  And since we have linear factors of multiplicity of 1, the graph will cross the x-axis.  Therefore, the intervals of x in which the graph is on or below the x-axis is

 

(-∞, -6]∪[-2, 4/3]

 

 

Now you just graph this interval on a number line.

Answer 2

To solve this, first figure out the endpoints by setting each factor =0. So they are -6,-2,4/3.  We know these are included in the solution bc it is less than or equals 0 so they are colored in circles. Then you place these points on the number line and pick points both to the right and left of each and see if the product of these factors are negative. I checked 5 (didn't work). 0 did bc the first factor was negative and the other 2 were positive. -4 didn't work and -7 did. 

 

So then the solution set would be

(-infinity,-6] U [-2,4/3].