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Chapter 7: Problem 10

Sketch the graph of the inequality. $$10 \geq y$$

Short Answer

Expert verified
The graph of this inequality is a solid horizontal line drawn at the point \(y = 10\) and the area below the line is shaded, which represents all points \(y\) with a value less than or equal to 10.

Step by step solution

01

Setup

The inequality given is \(10 \geq y\). We need to draw the line corresponding to \(y = 10\) on the graph.
02

Draw the line

Draw a horizontal line passing through the point \(y = 10\). This is the line that represents \(y = 10\). Because the inequality symbol includes 'equal to', the line should be solid \(not dashed\). This shows that points on the line are included in the solution.
03

Shading the graph

Since the inequality is \(10 \geq y\), which implies that y is less than or equal to 10, we need to shade below the line \(y = 10\). This shading represents the set of points that satisfy the inequality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality Graph
Inequality graphs visually represent the set of solutions for an inequality. Unlike equations, which typically result in a single, perfect line or curve, inequalities show ranges of values. They demonstrate which areas on the graph satisfy the inequality and which do not.
This is done by graphically plotting the equation as if it were an equal sign. For the inequality \(10 \geq y\), you begin by plotting \(y = 10\). In essence:
  • Convert the inequality to an equation to find the boundary line.
  • Consider the nature of the inequality (greater than, less than, etc.) to decide how to proceed with shading and line representation.
Using a graph like this helps in quickly identifying the region where the inequality holds true. Such visualizations are incredibly helpful in understanding the behavior of the inequality without manually substituting numerous values.
Shading Regions
Shading regions in a graph emphasizes which parts of the graph satisfy the inequality condition. Once you have your boundary line, the next important task is determining where to shade. For \(10 \geq y\), the inequality indicates all \(y\) values less than or equal to 10.
Here's how to decide on shading:
  • Examine the direction of the inequality sign. Since it's \(\geq\), we are interested in values that are less than or equal to 10.
  • Shade the region below the line \(y = 10\). Shading below means including points that have \(y\) values lower than 10.
  • Remember, the line itself is also part of the solution because of the 'equal鈥� component of the inequality.
Shading brings clarity and makes it easier to see the whole solution set quickly, showing which side of the boundary line encompasses the acceptable solutions.
Horizontal Line
In graphing, a horizontal line is a line where all points have the same \(y\)-coordinate. These lines are parallel to the x-axis and are straightforward to plot. When graphing inequalities like \(10 \geq y\), you would start by drawing the line \(y = 10\).
Key points about horizontal lines:
  • All points on a horizontal line have the same value of \(y\), such as 10 in our example.
  • Horizontal lines are represented by equations in the form \(y = c\), where \(c\) is a constant.
  • In inequalities with an 鈥榚qual to鈥� component (\(\geq\) or \(\leq\)), such lines are solid, conveying that points on the line also meet the inequality's requirements.
Understanding horizontal lines allows students to tackle a wider range of graphing inequalities efficiently, providing a clear understanding of the solutions they represent.

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Most popular questions from this chapter

Find the value of \(k\) such that the system of linear equations is inconsistent. $$\left\\{\begin{array}{l} 4 x-8 y=-3 \\ 2 x+k y=16 \end{array}\right.$$

A fruit grower raises crops \(A\) and \(B\). The yield is 300 bushels per acre for crop \(\mathrm{A}\) and 500 bushels per acre for crop B. Research and available resources indicate the following constraints. "The fruit grower has 150 acres of land for raising the crops. \(\cdot\) It takes 1 day to trim an acre of crop \(A\) and 2 days to trim an acre of crop \(\mathbf{B},\) and there are 240 days per year available for trimming. \(\cdot\) It takes 0.3 day to pick an acre of crop \(\mathrm{A}\) and 0.1 day to pick an acre of crop \(\mathbf{B},\) and there are 30 days per year available for picking. What is the optimal acreage for each fruit? What is the optimal yield?

Find values of \(x, y,\) and \(\lambda\) that satisfy the system. These systems arise in certain optimization problems in calculus, and \(\lambda\) is called a Lagrange multiplier. $$\left\\{\begin{aligned} 2+2 y+2 \lambda &=0 \\ 2 x+1+\lambda &=0 \\ 2 x+y-100 &=0 \end{aligned}\right.$$

System A system of pulleys is loaded with 128-pound and 32 -pound weights (see figure). The tensions \(t_{1}\) and \(t_{2}\) in the ropes and the acceleration \(a\) of the 32-pound weight are found by solving the system of equations $$ \left\\{\begin{aligned} t_{1}-2 t_{2}\quad\quad =0 \\ t_{1}\quad\quad -2 a=128 \\ \quad t_{2}+a =32 \end{aligned}\right. $$ where \(t_{1}\) and \(t_{2}\) are in pounds and \(a\) is in feet per second squared. Solve this system.

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