Question

Graph the solution set of each system of inequalities.y≤log⁡xy\le\log xy≥∣x−2∣y\ge\left|x-2\right|

Accepted Answer

Identify the domain and shape of each inequality. For the first inequality, $$y \leq \log x$$, note that the logarithm function $$\log x is $$defined only for $$x \u003e 0. $$The graph of $$y = \log x is a $$curve increasing slowly and passing through the point $$(1,0). $$For the second inequality, $$y \geq |x - 2|$$, recognize that $$y = |x - 2| is a V-$$shaped graph with its vertex at $$(2,0). $$The inequality $$y \geq |x - 2|$$ means the solution includes the region on or above this V-shaped graph. Graph the boundary lines first: plot $$y = \log x$$ for $$x \u003e 0$$ and $$y = |x - 2|$$ for all real $$x. $$Use a solid line for both since the inequalities include equality ($$\leq$$ and $$\geq). $$Determine the solution regions for each inequality separately. For $$y \leq \log x$$, shade the region below or on the curve $$y = \log x. $$For $$y \geq |x - 2|$$, shade the region above or on the V-shaped graph. Find the intersection of the two shaded regions from the previous step. The solution set of the system is where both shaded regions overlap. This overlapping region satisfies both inequalities simultaneously.

Graph the solution set of each system of inequalities.
$y \leq \log x$
$y \geq ∣ x - 2 ∣$

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Key Concepts

Graphing Inequalities

Logarithmic Functions

Absolute Value Functions

Graph the solution set of each system of inequalities.y≤logxy\(\le\]\log\) xy≥∣x−2∣y\(\ge\]\left\)|x-2\(\right\)|