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For each interval expressed in the number lines, give the interval notation and set-builder notation.

Type inf or infinity for $\infty\text{,}$ >= for $\geq\text{,}$ and <= for $\leq\text{.}$

In set-builder notation:

In interval notation:

In set-builder notation:

In interval notation:

In set-builder notation:

In interval notation:

Solution

Since all numbers less than or equal to $2$ are shaded, the set-builder notation is ${\{ x \mid x \le 2 \}}\text{.}$ The shaded interval “starts” at $-\infty$ and ends at $2$ (including $2$ ) so the interval notation is ${\left(-\infty ,2\right]}\text{.}$

Since all numbers less than to $2$ are shaded, the set-builder notation is ${\{ x \mid x < 2 \}}\text{.}$ The shaded interval “starts” at $-\infty$ and ends at $2$ (excluding $2$ ) so the interval notation is ${\left(-\infty ,2\right)}\text{.}$

Since all numbers greater than or equal to $2$ are shaded, the set-builder notation is ${\{ x \mid x \ge 2 \}}\text{.}$ The shaded interval starts at $2$ (including $2$ ) and “ends” at $\infty\text{,}$ so the interval notation is ${\left[2,\infty \right)}\text{.}$