Problem 39
Question
Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}y \leq \sqrt{3 x}+1 \\ y \geq x+1\end{array}\right.$$
Step-by-Step Solution
Verified Answer
The intersection of the regions defined by the two inequalities lies between the line \(y = x + 1\) and the curve \(y = \sqrt{3x} + 1\). This is the solution set of the system of inequalities.
1Step 1: Graph the first inequality
The equation \(y = \sqrt{3x} + 1\) represents a transformation of the basic square root function, shifted one unit vertically upward. The inequality \(y \leq \sqrt{3x} + 1\) then includes all points below this curve. Start by graphing the curve, and then show the region that represents all y-values that are less than or equal to this curve.
2Step 2: Graph the second inequality
The equation \(y = x + 1\) represents a line, with a slope of 1 and a y-intercept of 1. The inequality \(y \geq x + 1\) includes all points above this line. Draw this line on the same graph as the first inequality, and show the region represents all y-values that are greater than or equal to this line.
3Step 3: Identify the intersection of solutions
The solution set to the system of inequalities is the intersection of the solution sets of the individual inequalities. This will be the region that lies below the curve of the first inequality and above the line of the second inequality. Identify this region on your graph.
Key Concepts
Inequality GraphingSquare Root FunctionsLinear Functions
Inequality Graphing
Inequality graphing is an essential tool used in algebra to visually represent the solution sets of inequalities. Unlike equations, which show equality, inequalities show a range of values that are either greater than or less than a particular function on a graph.
When graphing an inequality, such as the textbook example with the inequalities \(y \leq \sqrt{3x} + 1\) and \(y \geq x + 1\), it is important to first graph the boundary lines, which in this case are \(y = \sqrt{3x} + 1\) and \(y = x + 1\). Then, to identify the solution sets, you generally shade the region that satisfies the inequality. For instance, with \(y \leq\) the graph shows the region below the curve, whereas with \(y \geq\) it shows the region above the line.
It's vital to use a dotted line for inequalities that are '<' or '>', which do not include the boundary itself, and a solid line for '\leq' or '\geq' inequalities which do include the boundary. Once both inequalities are graphed, the solution to the system of inequalities is where the shaded regions overlap, implying the set of coordinates that fulfill both conditions simultaneously.
When graphing an inequality, such as the textbook example with the inequalities \(y \leq \sqrt{3x} + 1\) and \(y \geq x + 1\), it is important to first graph the boundary lines, which in this case are \(y = \sqrt{3x} + 1\) and \(y = x + 1\). Then, to identify the solution sets, you generally shade the region that satisfies the inequality. For instance, with \(y \leq\) the graph shows the region below the curve, whereas with \(y \geq\) it shows the region above the line.
It's vital to use a dotted line for inequalities that are '<' or '>', which do not include the boundary itself, and a solid line for '\leq' or '\geq' inequalities which do include the boundary. Once both inequalities are graphed, the solution to the system of inequalities is where the shaded regions overlap, implying the set of coordinates that fulfill both conditions simultaneously.
Square Root Functions
Square root functions are a type of radical function that are especially relevant when working with quadratic relationships. The basic parent function of a square root graph is \(y = \sqrt{x}\), which forms a characteristic 'sideways parabola,' starting at the origin (0,0).
Transformations can occur to this basic function, such as horizontal shifts, vertical shifts, reflections, or even stretching and compressing the graph. For the given exercise, the function \(y = \sqrt{3x} + 1\) has been vertically shifted upwards by 1 unit. When graphing the inequality \(y \leq \sqrt{3x} + 1\), every point on the coordinate plane that lies below or on this transformed curve is considered as part of the solution set.
Understanding how to draw these functions and their transformations is critical in visualizing the relationship they describe and how they intersect with other types of functions, such as the linear functions in the system of inequalities.
Transformations can occur to this basic function, such as horizontal shifts, vertical shifts, reflections, or even stretching and compressing the graph. For the given exercise, the function \(y = \sqrt{3x} + 1\) has been vertically shifted upwards by 1 unit. When graphing the inequality \(y \leq \sqrt{3x} + 1\), every point on the coordinate plane that lies below or on this transformed curve is considered as part of the solution set.
Understanding how to draw these functions and their transformations is critical in visualizing the relationship they describe and how they intersect with other types of functions, such as the linear functions in the system of inequalities.
Linear Functions
Linear functions are the most fundamental type of function in algebra, with a standard form of \(y = mx + b\) where \(m\) represents the slope and \(b\) represents the y-intercept. These functions graph as straight lines and are a key concept in understanding algebraic relationships.
The exercise presented features a linear function in the inequality \(y \geq x + 1\), which indicates that all points above the line \(y = x + 1\) are included in the solution set for this particular inequality. In the context of the system of inequalities, recognizing how this linear function behaves is crucial for finding the region of intersection with the square root function.
In a graph, the line forms a slope of 1, which means for every unit you go right, you also go up by one unit. The y-intercept at 1 signifies where the line crosses the y-axis. Documenting these changes through graphing allows students to analyze and solve systems of linear inequalities effectively.
The exercise presented features a linear function in the inequality \(y \geq x + 1\), which indicates that all points above the line \(y = x + 1\) are included in the solution set for this particular inequality. In the context of the system of inequalities, recognizing how this linear function behaves is crucial for finding the region of intersection with the square root function.
In a graph, the line forms a slope of 1, which means for every unit you go right, you also go up by one unit. The y-intercept at 1 signifies where the line crosses the y-axis. Documenting these changes through graphing allows students to analyze and solve systems of linear inequalities effectively.
Other exercises in this chapter
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