Problem 6

Question

The graphs of linear inequalities are given next. For each, find three points that satisfy the inequality and three that are not in the solution set. $y-2 x \geq 0$

Step-by-Step Solution

Verified

Answer

Three points that satisfy the inequality $y\geq 2x$ are (0,0), (1,3), and (-1,2), while three points that do not satisfy the inequality are (1,1), (2,2), and (0,-1).

1Step 1: Rewrite the inequality in slope-intercept form

First, let's rewrite the given inequality, $y-2x\geq 0$, in slope-intercept form (i.e., $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept). $y-2x\geq 0$ Add $2x$ to both sides of the inequality: $y \geq 2x$ Now, the inequality is in slope-intercept form: $y \geq 2x$ The boundary line is given by the equation: $ y = 2x$

2Step 2: Find three points that satisfy the inequality

To find points that satisfy the inequality, we can choose points on or above the boundary line $y = 2x$. Let's pick three points and check if they satisfy the inequality. 1. Point (0,0): Since the inequality is true when $x=0$ and $y=0$, this point lies on the boundary line and satisfies the inequality. So, (0,0) is in the solution set. 2. Point (1,3): Plug in the values of $x=1$ and $y=3$ in the inequality: $3 \geq 2(1)$ $3 \geq 2$ The point (1,3) satisfies the inequality, so it is in the solution set. 3. Point (-1,2): Plug in the values of $x=-1$ and $y=2$ in the inequality: $2 \geq 2(-1)$ $2 \geq -2$ The point (-1,2) satisfies the inequality, so it is in the solution set.

3Step 3: Find three points that do not satisfy the inequality

Now, let's find points that do not satisfy the inequality. These points will be below the boundary line $y=2x$. 1. Point (1,1): Plug in the values of $x=1$ and $y=1$ in the inequality: $1 \nless 2(1)$ $1 \nless 2$ Since the inequality is not true, the point (1,1) is not in the solution set. 2. Point (2,2): Plug in the values of $x=2$ and $y=2$ in the inequality: $2 \nless 2(2)$ $2 \nless 4$ Since the inequality is not true, the point (2,2) is not in the solution set. 3. Point (0,-1): Plug in the values of $x=0$ and $y=-1$ in the inequality: $-1 \nless 2(0)$ $-1 \nless 0$ Since the inequality is not true, the point (0,-1) is not in the solution set. In conclusion, the points (0,0), (1,3), and (-1,2) satisfy the inequality $y\geq 2x$, while the points (1,1), (2,2), and (0,-1) do not.

Key Concepts

Slope-Intercept FormSolution SetBoundary Line

Slope-Intercept Form

Understanding the slope-intercept form is crucial in graphing linear equations and inequalities. The slope-intercept form of a linear equation is expressed as:
\[ y = mx + b \]
Here, $m$ represents the slope, which indicates the steepness and direction of the line. $b$ is the y-intercept, the point where the line crosses the y-axis.
To convert any linear inequality into this form makes it easier to visualize its graph. For instance, when the inequality $y - 2x \geq 0$ is rewritten as the slope-intercept form $y \geq 2x$ it conveys that the line has a slope of $2$ and passes through the origin ($0,0$).
This conversion helps in graphing because we know exactly how the line will look and how it divides the coordinate plane.

Solution Set

The solution set of a linear inequality consists of all points that satisfy the inequality. After rewriting the inequality $y \geq 2x$, the aim is to find all points that lie on or above the boundary line $y = 2x$.
To determine if a point is part of the solution set, substitute the coordinates into the inequality formula and check if the inequality holds true:

Example: $(0,0)$ satisfies $y \geq 2x$. Insert $x = 0$ and $y = 0$, leading to $0 \geq 0$. Thus, $(0,0)$ is part of the solution set.
Conversely, $(1,1)$ does not satisfy the inequality, since $1 ot\geq 2$.

By selecting various points and checking their validity with the inequality, you can distinguish which points are included or excluded in the solution set.

Boundary Line

The concept of a boundary line helps in visualizing linear inequalities. The boundary line, given by the equation $y = 2x$, is a crucial element because it separates the plane into regions that either satisfy or do not satisfy the inequality.
When graphing linear inequalities, the boundary line serves as a reference:

If the inequality uses a "greater than or equal to" ($\geq$) or "less than or equal to" ($\leq$) sign, the boundary line is solid, indicating the points on the line satisfy the inequality.
If the inequality uses a "greater than" ($>$) or "less than" ($<$) sign, the boundary line is dashed, meaning points on the line are not included in the solution set.

For the inequality $y \geq 2x$, the boundary line is solid. All points on or above this line satisfy the inequality. It’s essential to understand this division as it guides us in shading the correct region of the graph, representing the solution set.

Problem 6

Other exercises in this chapter

Write a system of linear equations in $x$ and $y$ represented by each augmented matrix. $$\left[\begin{array}{rr|r}1 & -1 & 6 \\\\-4 & 7 & 2\end{array}\righ

View solution

Problem 6

Graph each inequality on a number line and represent the sets of numbers using interval notation. $$-\frac{1}{4} \leq q \leq \frac{11}{4}$$