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# Inequalities

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1. How do you get the range of the inequality when you've got 'x is less than or equal to plus or minus 4' and 'x is less than or equal to plus or minus 2'? How do you know which values to put in the range together?
2. I think drawing a graph helps with these questions (and hence using a graphical calculator). You want abs(x^2 -10) is less than or equal to 6. So you'll want to draw y = abs(x^2 -10) and y=6, like so:

You want when the modulus graph is less than or equal to the line y=6, so I've highlighted the areas of the graph you want:

Attachment 545051545059

This shows that you have four possible solutions for this graph (as there are four points where the modulus graph and the line y=6 intersect). So in order to get these solutions, you're solving x^2 - 10 = 6 and -x^2 + 10 = 6, which gives you x = +- 4 and x = +- 2. So your inequalities are -4 =< x =< -2 (for the left side of the graph) and 2 =< x =< 4 (for the right side of the graph), as these are the areas of x when the modulus graph is below/equal to the line of y=6.

That's how we've been taught to solve these inequality questions anyhow. Hope this helped!
Attached Images

3. (Original post by sophiebookley)
I think drawing a graph helps with these questions (and hence using a graphical calculator). You want abs(x^2 -10) is less than or equal to 6. So you'll want to draw y = abs(x^2 -10) and y=6, like so:

You want when the modulus graph is less than or equal to the line y=6, so I've highlighted the areas of the graph you want:

Attachment 545051545059

This shows that you have four possible solutions for this graph (as there are four points where the modulus graph and the line y=6 intersect). So in order to get these solutions, you're solving x^2 - 10 = 6 and -x^2 + 10 = 6, which gives you x = +- 4 and x = +- 2. So your inequalities are -4 =< x =< -2 (for the left side of the graph) and 2 =< x =< 4 (for the right side of the graph), as these are the areas of x when the modulus graph is below/equal to the line of y=6.

That's how we've been taught to solve these inequality questions anyhow. Hope this helped!

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