Hi Mike, I find these Qs confusing and I lose valuable time trying to think my way through them (usually get them wrong anyway!) What’s a good stepwise approach? Thanks!

If 6 < |x-3| < 7 and x < 0, what is one possible value of |x| ?

I think it’s helpful on questions like this to plug in—to think in terms of actual numbers as much as possible. In this case, I’d look at the inequality, which says that |x-3| must be between 6 and 7, and I’d simplify that by saying that 6.5 is between 6 and 7, so why not just say |x-3|=6.5?

From there, you can say that x-3=6.5, in which case x=9.5, or x-3=-6.5, in which case x=-3.5. Since the question says that x<0, we must choose x=-3.5, which means |x|=3.5. So that’s what we grid.

Algebraically this is a bit trickier, but if you want to see a step-by-step solution, you have to begin by noting that removing the absolute value brackets from the original inequality range results in two possible inequality ranges:


Again, we know that x<0, so we only need to deal with the second inequality. Solve for x by adding 3 to each part:


So x must be between –3 and –4. That means the absolute value of x must be between 3 and 4.

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