Maths C3 functions

find solution to inequality |1/4x+3| ≥ 3

I've squared both sides and rearranged to get x^2/16 + 3/2x ≥ 0

where do i go from here?

Try factorising the left hand side, then consider which values of x can satisfy the inequality.

if i times both sides by 16 to get x^2 + 24x ≥ 0

where do i go from there??

You can factorise the left hand side into:

x(x+24) ≥ 0

Now, you can either draw the graph of the function of y=x(x+24) and see where
y≥0 to answer this question, or just consider when this function is positive.

The function x(x+24) is a product of two numbers. So, for the answer to be ≥0, either one of the two numbers is 0, both the numbers are positive or both the numbers are negative. By looking at the function, it is inherently obvious that x=0 and x=-24 are both solutions.

This gives us our two critical values for our inequality. If you think about a graph for a quadratic function, there are two possible shapes. An x^2 graph dips below the x axis (0≥y) between the critical values, and is positive elsewhere, whereas a -x^2 graph dips above the x-axis (y≥0) between the critical values, and is negative elsewhere.

Our function is a positive x^2, so we know it is ≥0 outside of the critical values. So the answer is:

x≥0 or -24≥x

find solution to inequality |1/4x+3| ≥ 3

I've squared both sides and rearranged to get x^2/16 + 3/2x ≥ 0

where do i go from here?

Just thought I would add this: Be very careful when squaring inequalities. It is generally safer to either sketch a graph and solve for intersection points, or multiply by the square of the denominator.