Use algebraic and/or graphical methods to solve the inequalities

Question

A. x⁴-4x² ≤ 0

 

 

B. x/x-1 ≤ 2x+ 1/x-1

Answer 1

Hi Mark! This problem is pretty tough to explain through typing, but I will give it a go!

 

A. Step 1: attempt to factor the left side of the inequality. You should end up with three factors. (Factor out gcf, then use difference of squares rule).

 

Step 2: Set each factor you obtained equal to zero, and solve for x. These three numbers are considered your critical points, and outline the four possible solution intervals.

 

Step 3: Next, you want to select four test points that fall within the four possible solution intervals and plug those test points into the original inequality one at a time. Note the test points where you achieve a true statement, such as -1<0.

 

Step 4: Write the solution intervals based on the the test points that lead to true inequalities in step 3. This is most often done in interval notation.

 

Here is an example similar to A:

x³-16x<0

 

Solution:

Step 1: x³-16x=x(x²-16)=x(x+4)(x-4)<0

 

Step 2: x=0, x+4=0, x-4=0

solve for x: x=-4,0,4

POSSIBLE solution intervals: (-inf, -4), (-4,0), (0,4), (4,inf) *Note that your problem will contain some brackets because your inequality sign is less than or equal to)

 

Step 3: Test point for (-inf, -4): -5 *could have chosen any number within the interval.

Test point for (-4,0): -1

Test point for (0,4): 1

Test point for (4,inf): 5

now plug in each test point separately into original inequality and note ones that result in a true statement.

x=-5 : (-5)³-16(-5)<0   -45<0 TRUE

x=-1 : (-1)³-16(-1)<0   15<0 FALSE

x=1 : (1)³-16(1)<0.      -15<0 TRUE

X=5 : (5)³-16(5)<0.      45<0 FALSE

 

Step 4: Solution: (-inf,-4)U(0,4) 

*note that since there is at least one number between my intervals, I keep them separate and connect them with the union symbol U. This is most often, BUT NOT ALWAYS the case.

 

I hope this helps! Send me an email if you are confused at any of these steps as they pertain to your problem.

 

Shannon L.

 

PS. I'll write another post and help you with B.