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Graphing inequalities involves representing solutions on a coordinate plane. When you graph an inequality like \(y \leq \log x\), you first graph the boundary equation, \(y = \log x\). This shows you the curve that serves as the limit for your inequality. Remember that logarithmic functions have a vertical asymptote at \(x = 0\), which means the graph will never actually touch or cross this line. This helps you to visualize the boundary effectively.
Once the boundary is plotted, the inequality \(\leq\) indicates that the solution set includes the area below the boundary line. Use shading techniques to denote this entire region on your graph. Similarly, the inequality \(y \geq |x-2|\) involves graphing the absolute value function \(y = |x-2|\), a 'V'-shaped graph. Again, shading above this 'V'-line helps identify the solution region, including the boundary itself.

• Graphs are done on a coordinate plane.
• Familiarize yourself with the shape of each function involved.
• Shade the correct regions depending on the inequality symbols.

Understanding these concepts helps in visualizing how different regions on the graph represent solutions to the inequalities.

Logarithmic functions are a fundamental part of algebra and are often encountered in graphing exercises involving inequalities. The function \(y = \log x\) is a classic example of a logarithmic function, showing the relationship where \(y\) represents the power to which a base (commonly 10 in base-10 logarithms) must be raised to yield \(x\). This function is only defined for positive values of \(x\), and it features a vertical asymptote at \(x = 0\).
When graphing the logarithmic function \(y = \log x\), recognize some key characteristics:

• The graph passes through the point \( (1, 0) \), since \(\log(1) = 0\).
• It gradually increases, moving to the right as \(x\) increases.
• The curve is never horizontal and doesn't touch the vertical asymptote.

This understanding is crucial when solving inequalities such as \(y \leq \log x\). Recognize that your shading should include the points on the graph where \(y\) is equal to \(\log x\) and below, showcasing a comprehensive solution.

Absolute value functions, such as \(y = |x-2|\), create a distinct 'V' shape on a graph. This type of graph is pivotal when studying systems of inequalities, because it demonstrates different behaviors depending on the value of \(x\).
The vertex of the 'V' is at \( (2, 0) \), which is the point where the graph changes direction. For \(x \geq 2\), the function follows the line \(y = x-2\), and for \(x < 2\), the graph mirrors this over the y-axis, following \(y = -(x-2)\).

• The vertex represents the minimum point of the graph.
• The graph is symmetric around the vertex.

In a situation where you graph \(y \geq |x-2|\), it signifies shading the region above and including the 'V' itself. This includes all points that satisfy the condition \(y = |x-2|\) and those above it, portraying the full solution set for the inequality. Understanding how to navigate through absolute value functions is key to solving and graphing more complex systems of inequalities efficiently.