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Chapter 3: Problem 5

On different grids, graph each inequality (shading in the appropriate area) and then determine whether or not the origin, the point \((0,0),\) satisfies the inequality. a. \(-2 x+6 y<4\) b. \(x \geq 3\) c. \(y>3 x-7\) d. \(y-3>x+2\)

Short Answer

Expert verified
a. Satisfies; b. Does not satisfy; c. Satisfies; d. Does not satisfy.

Step by step solution

01

Title - Graph the inequality \(-2x + 6y < 4\)

First, rewrite the inequality in slope-intercept form, solving for y: \[-2x + 6y < 4\] Add 2x to both sides: \[6y < 2x + 4\] Divide by 6: \[y < \frac{1}{3}x + \frac{2}{3}\] Graph the boundary line \(y = \frac{1}{3}x + \frac{2}{3}\) with a dashed line (since it鈥檚 a strict inequality). Shade the region below the line. Check the origin by substituting \(0,0\): \(-2(0) + 6(0) < 4\)\[0 < 4\] The origin satisfies the inequality.
02

Title - Graph the inequality \(x \geq 3\)

Graph the boundary line \x = 3\ with a solid line (since it鈥檚 an inclusive inequality). Shade the region to the right of the line. Check the origin by substituting \(0,0\): \[0 \geq 3\] This is false, so the origin does not satisfy the inequality.
03

Title - Graph the inequality \y > 3x - 7\)

Rewrite the inequality: \[y > 3x - 7\] Graph the boundary line \(y = 3x - 7\) with a dashed line (since it鈥檚 a strict inequality), and shade the region above the line. Check the origin by substituting \(0,0\): \[0 > 3(0) - 7\] \[0 > -7\] The origin satisfies the inequality.
04

Title - Graph the inequality \y - 3 > x + 2\)

Rewrite the inequality: \[y - 3 > x + 2\] Add 3 to both sides: \[y > x + 5\] Graph the boundary line \(y = x + 5\) with a dashed line (since it鈥檚 a strict inequality), and shade the region above the line. Check the origin by substituting \(0,0\): \[0 - 3 > 0 + 2\] \[-3 > 2\] This is false, so the origin does not satisfy the inequality.

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Key Concepts

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

slope-intercept form
The slope-intercept form of a linear equation is a fundamental concept when graphing inequalities. This form is expressed as: \[ y = mx + b \] where
  • \( m \) is the slope of the line, representing the rate of change
  • \( b \) is the y-intercept, where the line crosses the y-axis.
For instance, consider the inequality \(-2x + 6y < 4\). By rearranging it into slope-intercept form, we get: \[y < \frac{1}{3} x + \frac{2}{3} \] This makes graphing simpler as you can quickly plot the y-intercept (\( \frac{2}{3} \) in this case) and use the slope (\( \frac{1}{3} \)) to plot other points along the line. This method is foundational, allowing you to graph any linear inequality with ease.
region shading
Region shading is key to visualizing solutions of inequalities on a graph. After plotting the boundary line, decide which region to shade based on the inequality's direction:
  • If your inequality uses '<' or '鈮�', shade below the line.
  • If it uses '>' or '鈮�', shade above the line.
This shaded region represents all possible solutions to the inequality. For example, for \( y < \frac{1}{3} x + \frac{2}{3} \), shade below the line, indicating that all points in this area satisfy the inequality. Shading correctly will help you understand the range of values that meet the inequality's condition.
boundary lines
Boundary lines are the lines you draw based on the linear equation derived from the inequality. There are key differences depending on the inequality type:
  • For strict inequalities ('<', '>'), use a dashed line indicating points on the line are not included in the solution.
  • For inclusive inequalities ('鈮�', '鈮�'), use a solid line, showing points on the line are part of the solution set.
For instance, in \(-2x + 6y < 4\), after converting to slope-intercept form \( y < \frac{1}{3} x + \frac{2}{3} \), we use a dashed line to graph \( y = \frac{1}{3} x + \frac{2}{3} \). This visual differentiation helps indicate whether boundary points are solutions or not.
inequality verification
Verifying inequality solutions involves checking if specific points satisfy the inequality. The origin (\(0,0\)) is often checked for convenience. Substituting \(0,0\) into the inequality helps determine if it's a solution:
  • If true, the origin is part of the solution set.
  • If false, it is not.
For instance, for \(-2x + 6y < 4\), substituting \(0,0\) results in \(0 < 4\), which is true, indicating the origin satisfies this inequality. This method quickly verifies whether particular points lie within the solution's region, providing a practical way to test other points as needed.

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Most popular questions from this chapter

The supply and demand equations for a particular bicycle model relate price per bicycle, \(p\) (in dollars) and \(q,\) the number of units (in thousands). The two equations are $$ p=250+40 q \quad \text { Supply } $$ $$ p=510-25 q $$ Demand a. Sketch both equations on the same graph. On your graph identify the supply equation and the demand equation. b. Find the equilibrium point and interpret its meaning.

Construct a sketch of each system by hand and then estimate the solution(s) to the system (if any). $$ \begin{array}{ll} \text { a. } x+2 y=1 & \text { b. } x+y=9 \\ x+4 y=3 & 2 x-3 y=-2 \end{array} $$

(Graphing program recommended.) The blood alcohol concentration (BAC) limits for drivers vary from state to state, but for drivers under the age of 21 it is commonly set at 0.02. This level (depending upon weight and medication levels) may be exceeded after drinking only one 12 -oz can of beer. The formula $$ N=6.4+0.0625(W-100) $$ gives the number of ounces of beer, \(N,\) that will produce a BAC legal limit of 0.02 for an average person of weight \(W\). The formula works best for drivers weighing between 100 and \(200 \mathrm{lb}\). a. Write an inequality that describes the condition of too much blood alcohol for drivers under 21 to legally drive. b. Graph the corresponding equation and label the areas that represent legally safe to drive, and not legally safe to drive conditions. c. How many ounces of beer is it legally safe for a 100 -lb person to consume? A 150-lb person? A 200-lb person? d. Simplify your formula in part (b) to the standard \(y=m x+b\) form. e. Would you say that " 6 oz of beer +1 oz for every \(20 \mathrm{lb}\) over \(100 \mathrm{lb}\) " is a legally safe rule to follow?

Determine whether or not the point (-1,3) satisfies the inequality. a. \(x-3 y>6\) b. \(x<3\) c. \(y \leq 3\) d. \(y \leq-\frac{1}{2} x+3\)

Solve the following system of three equations in three variables, using the steps outlined below: $$ \begin{aligned} 2 x+3 y-z &=11 \\ 5 x-2 y+3 z &=35 \\ x-5 y+4 z &=18 \end{aligned} $$ a. Use Equations (1) and (2) to eliminate one variable, creating a new Equation (4) in two variables. b. Use Equations (1) and (3) to eliminate the same variable as in part (a). You should end up with a new Equation (5) that has the same variables as Equation (4). c. Equations (4) and (5) represent a system of two equations in two variables. Solve the system. d. Find the corresponding value for the variable eliminated in part (a). e. Check your work by making sure your solution works in all three original equations.
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