Question

PLEASE PLEASE HELP 100 POINTS

Answer 1

x < 4 or 6 < x

Explanation:

You want the compound inequality that describes the number line graph with open circles at 4 and 6, and shading everywhere except between them.

Inequalities

The left line segment has all values less than 4 shaded. The corresponding inequality is ...

x < 4

The right line segment has all values greater than 6 shaded. The corresponding inequality is ...

x > 6

These do no overlap, so the compound inequality uses "or" to combine them:

x < 4 or x > 6

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Additional comment

Sometimes it is helpful to have the relationship between the shading and the limit values a little more obvious in the inequality. That will be the case if you always use left-pointing inequality symbols: < or ≤. By writing the compound inequality as ...

x < 4 or 6 < x

we see that shading is outside (left or right of) the interval [4, 6].

The symbol without the "or equal to" bar is used when the circle is an open circle. A filled dot would signify the limit value is part of the interval of interest, and the ≤ (or ≥) symbol would be used.