Given h(x) = -x + 1, solve for x when h(x) = 0.

Average velocity = 0 gives the horizontal component of the launch velocity .

What does average velocity mean?

The difference between the change in position or displacement (x) and the time periods (t) during which the displacement happens is known as average velocity.

                     Depending on how the displacement is displaced, the average velocity may be positive or negative. Meters per second (m/s or ms-1) is the standard international unit for average velocity.

Typical flight of a projectile is as shown in the picture above.

In the problem it is given that initial velocity V₀ at an angle θ

above the horizontal. As such inn the picture U=  V₀

This velocity can be resolved into its  x and y components.

Component along x axis, and

Component along y  axis = V₀ Sin θ

Let t be time of flight.

    Average velocity = Displacement/time of flight

  It is given that "It lands at the same level from which it was launched", means that displacement in the y axis is = 0  

      Average velocity  = 0/t  = 0  ............1

cos  θ component.

 Average velocity  =    V₀  cos  θ ............ 2

Average velocity we need to add both vectors along x and y directions

. In this instant it is simple as one of the vectors is  

                        Average velocity  = 0

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To solve a 3×3 augmented matrix use the method of elementary row operations.

What is a matrix?

A matrix is a rectangular array or table with numbers or other objects arranged in rows and columns. Matrices is the plural version of matrix. The number of columns and rows is unlimited. Matrix operations include addition, scalar multiplication, multiplication, transposition, and many others.

An augmented matrix for a system of equations is a matrix of numbers where each column contains all the coefficients for a single variable and each row represents the constants from one equation (both the coefficients and the constant on the other side of the equal sign).

The system of equations are - x - 2y + 3z = 7, 2x + y + z = 4, -3x + 2y -2z = -10

Here is the augmented matrix for this system.

[tex]\left[\begin{array}{rrr|r}1 & -2 & 3 & 7 \\2 & 1 & 1 & 4 \\-3 & 2 & -2 & -10\end{array}\right][/tex]

This matrix can be solved using the method of elementary row operations.

Interchange Two Rows. With this operation interchange all the entries in row  i and row j. The notation used here is Ri ↔ Rj.

[tex]\left[\begin{array}{rrr|r}1 & -2 & 3 & 7 \\2 & 1 & 1 & 4 \\-3 & 2 & -2 & -10\end{array}\right] \stackrel{R_1 \leftrightarrow R_3}{\rightarrow}\left[\begin{array}{rrr|r}-3 & 2 & -2 & -10 \\2 & 1 & 1 & 4 \\1 & -2 & 3 & 7\end{array}\right][/tex]

Multiply a Row by a Constant. In this operation multiply row i by a constant c and the notation will be cRi.

[tex]\left[\begin{array}{rrr|r}1 & -2 & 3 & 7 \\2 & 1 & 1 & 4 \\-3 & 2 & -2 & -10\end{array}\right] \stackrel{-4 R_3}{\rightarrow}\left[\begin{array}{rrr|r}1 & -2 & 3 & 7 \\2 & 1 & 1 & 4 \\12 & -8 & 8 & 40\end{array}\right][/tex]

Add a Multiple of a Row to Another Row.

Row i will be replaced in this procedure with row i times a constant c plus row j. Ri + cRi → Rj is the notation for this operation. This procedure involves taking an input from row i multiplying it by c, adding the equivalent value from row j, and then returning the result to row i.

[tex]\left[\begin{array}{rrr|r}1 & -2 & 3 & 7 \\2 & 1 & 1 & 4 \\-3 & 2 & -2 & -10\end{array}\right] \begin{gathered}R_3-4 R_1 \rightarrow R_3 \\\rightarrow\end{gathered}\left[\begin{array}{rrr|r}1 & -2 & 3 & 7 \\2 & 1 & 1 & 4 \\-7 & 10 & -14 & -38\end{array}\right][/tex]

Let’s go through the individual computation to make sure you followed this.

-3 - 4(1) = -7

2 - 4(-2) = 10

-2 - 4(3) = -14

-10 - 4(7) = -38

Therefore, the matrix is solved using row operations.

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With the identification of plans, angles, Rectangles, and segments, you can find the shoe box in the attachment.

Every point on a Plano has the same level because it is a space with only two dimensions.

The ángulos are a component of a plan that is created from two recitals with a common vertex. In the case of our shoe box, all angles are right-angled, despite the perspective appearing to be greater or smaller than 90 degrees.

Semirrectas are rectus with a known beginning but no known end. We can extend two segments in the shoe box so that they become semi-rectangles.

The segments are straight lines with clearly defined beginnings and end. All of the lines that form in the shoe box are segments.

The following is identified in the drawing.

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The Full question:

Consider a traditional and illustrated shoe box. This illustration indicates or names the following. ​

The value of integral ∫(0,8)  [4f(x) + 6g(x)] dx on the interval of (0,8) is 264

when ∫(0,8) f(x) dx = 39 and  ∫(0,8)g(x) dx =18.

Integration is a method of adding or summing up the parts to find the whole. It is a reverse process of differentiation, where we reduce the functions into parts.

Given that,

∫(0,8) f(x) dx = 39 and  ∫(0,8)g(x) dx =18

∫(0,8)  [4f(x) + 6g(x)] dx

Apply linearity rule of integration,

= ∫(0,8) 4f(x) dx + ∫(0,8) 6g(x) dx

=4 ∫(0,8) f(x) dx + 6∫(0,8) g(x) dx

= 4(39) + 6(18)

= 156 + 108

= 264

therefore, the value of integral ∫(0,8)  [4f(x) + 6g(x)] dx on the interval of (0,8) is 264

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For the velocity function v(t) = t³ - 3t² + 12t + 4, with interval 0 ≤ t ≤ 3, the maximum acceleration is option D: 21 m/s².

What is velocity?

The pace at which an object's position changes in relation to a frame of reference and time is what is meant by velocity. Although it may appear sophisticated, velocity is just the act of moving quickly in one direction. Since it is a vector quantity, the definition of velocity requires both magnitude (speed) and direction.

The velocity function is v(t) = t³ - 3t² + 12t + 4.

The interval is given as - [0,3]

Maximize the function to obtain the value of t.

v'(t) = a(t) = 3t² - 6t + 12

Maximize it again -

a'(t) = 6t - 6 = 0

6t = 0 + 6

t = 6/6

t = 1

Now there are three critical points for t = {0,1,3}

1 and 3 are endpoints of the interval [0,3].

To get the maximum acceleration, plug in the values of t in the equation .

First substitute the value t = 0 -

a(0) = 3t² - 6t + 12

a(0) = 3(0)² - 6(0) + 12

a(0) = 12

Now substitute the value t = 1 -

a(1) = 3t² - 6t + 12

a(1) = 3(1)² - 6(1) + 12

a(1) = 3 - 6 + 12

a(1) = 15 - 6

a(1) = 9

Now substitute the value t = 3 -

a(3) = 3t² - 6t + 12

a(3) = 3(3)² - 6(3) + 12

a(3) = 27 - 18 + 12

a(3) = 39 - 18

a(3) = 21

So, the maximum acceleration is 21 m/s² when t =3.

Therefore, the maximum acceleration is 21 m/s².

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The largest possible value of p can be determined by analysing the ratio of the volumes of the two rectangular boxes.

The volume of the first box is V1 = m*p*n and the volume of the second box is [tex]$V2 = (m2)\cdot (n2)\cdot (p2)$[/tex]. Therefore, we can set up the following equation to solve for p:

[tex]$\frac{V1}{V2} = \frac{mnp}{(m2)(n2)(p2)} = \frac{1}{2}$[/tex]

Solving for p gives us the following:

[tex]$p = \sqrt[3]{\frac{2(m2)(n2)}{mn}}$[/tex]

Since m, n, and p must all be integers, the largest possible value of p is the largest integer such that

[tex]$p \le \sqrt[3]{\frac{2(m2)(n2)}{mn}}$.[/tex]

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Answer:

X = 2

Step-by-step explanation:

5x + 10 = 2x + 16

3x + 10 = 16

3x = 6

x = 2

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The coefficients in the linear combination are determined by the initial or boundary conditions of the problem.

What is the differential equations?

A differential equation is an equation that relates an unknown function to its derivatives. It describes the behavior of a physical, biological, or engineering system in terms of changes in variables over time.

A fundamental solution in differential equations is a particular solution to a differential equation that contains arbitrary constants. It is called "fundamental" because it is a building block for constructing more general solutions to the equation. The general solution to a differential equation can be found by adding a linear combination of several fundamental solutions. The coefficients in the linear combination are determined by the initial or boundary conditions of the problem.

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Answer:

There are 5 ducks

Step-by-step explanation:

2 ducks behind

1 duck in the middle

2 ducks infront

The equation would be written in exponential form as follows: [tex]$$\ln(m) = n \Rightarrow m = e^n$$[/tex]

The equation ln(m)=n can be rewritten in exponential form as m=e^n. This can be seen by taking the natural logarithm (ln) of both sides of the equation. The natural logarithm of m is equal to n, so we have ln(m)=n. Applying the exponential function, e^x, to both sides of the equation gives us m=e^n. This can be further understood by calculating the exponential of both sides.

For example, if n = 1, then ln(m)=1, so m=e^1. Applying the exponential function, e^x, to both sides of the equation, we have m=e^1. Calculating e^1 gives us m=e^1=2.718. Thus, ln(m)=1 can be rewritten in exponential form as m=2.718.

The equation in Latex would be written as follows: [tex]$$\ln(m) = n \Rightarrow m = e^n$$[/tex]

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The vector parametric equations that describe the helical path you take from the point (28, 0, 0) to the point (28, 0, 40) as you make 4 revolutions during the time interval 0 ≤ t ≤ 1 are:

x(t) = 28 + 18 cos(πt / 15)

y(t) = 18 sin(πt / 15)

z(t) = 50t

To find a vector parametric equation for the helical path, we need to describe the position of a point on the spiral staircase as it moves from (28, 0, 0) to (28, 0, 40) while making 4 revolutions during the time interval 0 ≤ t ≤ 1.

Let's define the following parameters:

R: Radius of the spiral staircase = 18 feet

H: Height of the spiral staircase = 50 feet

N: Number of revolutions during the time interval = 4

T: Total time taken to complete N revolutions = 1 hour (or 60 minutes)

The parametric equations for the helical path can be given as follows:

x(t) = 28 + R × cos(2πNt/T)

y(t) = R × sin(2πNt/T)

z(t) = H × t

Where:

x(t), y(t), z(t) are the coordinates of the point on the helical path at time t.

R × cos(2πNt/T) and R × sin(2πNt/T) describe the circular motion of the point in the xy-plane as it makes N revolutions over the time interval.

H × t describes the linear motion of the point along the z-axis.

Now, let's plug in the given values:

R = 18 feet

H = 50 feet

N = 4

T = 60 minutes

And simplify the equations:

x(t) = 28 + 18 × cos(2π × 4t / 60)

y(t) = 18 × sin(2π × 4t / 60)

z(t) = 50 × t

Simplifying further:

x(t) = 28 + 18 × cos(π × t / 15)

y(t) = 18 × sin(π × t / 15)

z(t) = 50 × t

These equations describe the helical path you take from the point (28, 0, 0) to the point (28, 0, 40) as you make 4 revolutions during the time interval 0 ≤ t ≤ 1.

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The percentage score is 50%, for instance, if you get a test score of 50 out of 100. This can be calculated as follows: Percentage is (50/100) × 100, or 50. Proportion: A proportion is a statement indicating the equality of two ratios.

Two ratios are equal if they are equal by a certain proportion. Simply divide one part by the whole to get the proportion's ratio.

While percentage expresses a ratio or fraction with a constant denominator of 100, proportion expresses the equivalent of two ratios or fractions.

In order to express a component of a total as a fraction or a decimal, percentages and proportions are two related ideas. Here is how to determine each of them:

One technique to express a number as a fraction of 100 is to use a percentage. You multiply the part by 100 and divide by the whole to get a percentage. The formula to determine a percentage is as follows:

Percentage is equal to (Part/Whole) × 100.

To obtain the ratio and calculate a proportion, divide one part by the total. If you wish to know how many apples make up 25% of a collection of 20 apples, for instance, you can set up a proportion as follows:

Part / Whole = Percentage / 100

Part / 20 = 25 / 100

Part = (25 / 100) × 20 = 5

So, 25% of the collection is 5 apples.

In general, you can use these formulas to calculate percentages and proportions based on the information you have. The key is to always make sure you have the correct units and understand what the whole and part represent.

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The probability of selecting a $10 dollar bill or a $5 dollar bill would be = 13/23

Probability is defined as the expression that can be used to represent the possible outcome of an event which may likely occur or not.

The quantity of $1 bill = 10

The quantity of $5 bill = 10

The quantity of $10 bill = 3

The sum total of bills in the box = 23

The probability of choosing a 5 or 10 bills;

= 10+3/23

= 13/23

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Answer:

53 1/4

Step-by-step explanation:

55 5/6  - 1 1/3 - 1 1/4  Rewrite with a common denominator

55 10/12 - 1 4/12 - 1 3/12

55 10/12 - 1 4/12 = 54 6/12

54 6/12 - 1 3/12

53 3/12

53 1/4

Answer:

6

Step-by-step explanation:

If 5r+25=30, r=1. Therefore, r+5 would be 6.

Answer:

r has to be equal to 1, because 5 + 25 = 30, so r would be equal to 1 because 1 * 5 = 5 therefore the value will not change.

Now that we know that r = 1, the answer to r + 5 =

1 + 5 = 6

Step-by-step explanation:

Hope it helps! =D

P ↔ Q is a tautology because if P and Q are equivalent propositional forms, they have the same truth value in every possible interpretation, making the bi-conditional proposition P ↔ Q always true.

A tautology is a proposition that is always true, regardless of the truth values of its component propositions.

When two propositions, P and Q, are equivalent, it means that they have the same truth value in every possible interpretation. That is, P and Q are logically equivalent.

Therefore, if P and Q are equivalent propositional forms, then P ↔ Q, the bi-conditional proposition, is a tautology. This is because the truth value of P ↔ Q will always be true, as both P and Q have the same truth value in every possible interpretation.

This can be shown through the truth table for P ↔ Q. The bi-conditional proposition is true if and only if both P and Q have the same truth value. If P and Q are equivalent, then they will always have the same truth value, and so P ↔ Q will always be true.

In general, for any equivalent propositions P and Q, P ↔ Q will always be a tautology.

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This proof by induction shows that n + 1 log n = 2 for all positive integers n. This holds true since S(1) = 2, and S(k + 1) = 2 for any integer k ≥ 1.

Proof:

Let S(n) = n + 1 log n

We will prove S(n) = 2 by induction.

Base Case:

Let n = 1. Then S(1) = 1 + 1(log 1) = 1 + 0 = 1 = 2.

Inductive Step:

Assume S(k) = 2 for some arbitrary integer k ≥ 1. We must show that S(k + 1) = 2.

S(k + 1) = (k + 1) + 1(log(k + 1))

= k + 1 + 1(log k + log 1)

= k + 1 + 1(log k + 0)

= k + 1 + 1(log k)

= k + 1 + log k

= 2 + log k

= 2 (by induction hypothesis)

Therefore, S(n) = 2 for all positive integers n.

This proof by induction shows that n + 1 log n = 2 for all positive integers n. This holds true since S(1) = 2, and S(k + 1) = 2 for any integer k ≥ 1.

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The total number of different committees possible is 6 * 3 * 2 * 56 = 3456.

If 2 of the men refuse to serve together, then we have to choose the first man for the committee first and then choose the other two men one by one, making sure that the two men who refuse to serve together are not chosen together.

There are 6 possible choices for the first man. After the first man is chosen, there are 4 remaining men, and we have to choose 2 more men from these 4. However, one of these 2 men must be the one who refused to serve with the first man. So, there are only 3 possible choices for the second man. After the second man is chosen, there are only 2 possible choices for the third man.

Next, we have to choose 3 women from 8. There are C(8,3) = 56 ways to do this.

So, the total number of different committees possible is 6 * 3 * 2 * 56 = 3456.

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The total number of different committees possible is 6 * 3 * 2 * 56 = 3456.

If 2 of the men refuse to serve together, then we have to choose the first man for the committee first and then choose the other two men one by one, making sure that the two men who refuse to serve together are not chosen together.

There are 6 possible choices for the first man. After the first man is chosen, there are 4 remaining men, and we have to choose 2 more men from these 4. However, one of these 2 men must be the one who refused to serve with the first man. So, there are only 3 possible choices for the second man. After the second man is chosen, there are only 2 possible choices for the third man.

Next, we have to choose 3 women from 8. There are C(8,3) = 56 ways to do this.

So, the total number of different committees possible is 6 * 3 * 2 * 56 = 3456.

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The distance between David's house and school is definitely greater than 8.5miles as his school is more than 8.5 miles.

The answer must not be 8 , because as David stays 8.5 miles distance more from his school. An Inequality for a is a>8.5.

The illustration of two expressions by inequal symbol is known as inequality mathematical statement in algebra. It has non equal expressions on both sides. The inequality shows the values on the left side should be bigger or smaller than the expression on the right. The relationships between two algebraic expressions that are expressed using inequality symbols are literal inequalities. An algebraic expression is an expression built up from constant algebraic variables, numbers and the operators.

So the distance between David's house and school must be the value greater than 8.5

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Answer: 24 feet

Step-by-step explanation:

Perimeter: 2(length+width)

2(7+5)

14+10=24

Answer:

Sunrise; $35

Step-by-step explanation:

Hope it helps! =D

Ji-Yoon's loading speed is 6 trucks per hour.

From the question, we have the following parameters that can be used in our computation:

Ji-Yoon loaded 84 Trucks in 14 hours

Ji-Yoon's loading speed an be calculated by dividing the total number of trucks loaded (84) by the number of hours it took to load them (14):

This is repesented as

84/14 = 6

Hence, the speed is 6 trucks per hour

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The correct answer fort parallel angles is x = 33° and y = 10°.

The angles created when a transversal intersects two parallel lines are known as corresponding angles. Opening and shutting a lunchbox, completing a Rubik's cube, and an infinite stretch of parallel train lines are common examples of identical angles.

Angle 3y + 20° and angle 5y are parallel angles.

3y + 20° = 5y

2y =  20°

y = 10°

Angles 2x - 16° and 3y + 20° are in direct opposition to one another.

3y + 20° =  2x - 16°

Put the value of angle y in the equation.

3(10) + 20 = 2x - 16

30 + 20 = 2x - 16

50 = 2x - 16

2x = 66

x = 33

As a result, the angles are 33° for x and 10° for y.

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Question:

Find the value of x for which l is parallel to m. The diagram is not to scale. Lines l and m are parallel.

Based on the sample data, there is a 95% chance that the true proportion of all smartphone users who would admit to walking into something or someone while texting on their cell phone is between 0.199 and 0.271.

A 95% confidence interval for the proportion of all smartphone users who would admit to walking into something or someone while texting on their cell phone can be calculated as follows:

Let p be the true proportion of all smartphone users who would admit to walking into something or someone while texting on their cell phone. Based on the sample of 750 respondents, the point estimate of p is 176/750 = 0.235.

Using the normal approximation to the binomial distribution, the standard error of the point estimate is estimated as [tex]\sqrt{ (p(1-p)/n)}[/tex], where n = 750. This gives us a standard error of [tex]\sqrt{(0.235 * 0.765/750)}[/tex] = 0.018.

A 95% confidence interval for the true proportion p is then given by the point estimate plus or minus 1.96 times the standard error. This gives us the following interval:

0.235 - 1.96 * 0.018 <= p <= 0.235 + 1.96 * 0.018

0.199 <= p <= 0.271

So, we are 95% confident that the true proportion of all smartphone users who would admit to walking into something or someone while texting on their cell phone is between 0.199 and 0.271.

Interpretation: Based on the sample data, there is a 95% chance that the true proportion of all smartphone users who would admit to walking into something or someone while texting on their cell phone is between 0.199 and 0.271.

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log9 72 to four decimal places is 2.8594

To approximate log9 72 using common logarithms, we can use logarithmic properties and logarithmic tables.

First, we can rewrite 72 as [tex]9^3[/tex] to find the exponent that gives us 72:

log9 72 = log9 ([tex]9^3[/tex]) = 3

Now, we can use logarithmic tables or a calculator to find the common logarithm of 9, which is 0.954243:

log10 9 = 0.954243

Finally, we can divide the result by the common logarithm of 10 to find the logarithm to base 9:

log9 72 = (1/log10 9) * log10 72 = 0.954243 * log10 72 ≈ 2.859437

log9 72 to four decimal places is 2.8594.

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Ken borrowed P8,077.84 from Haguel for a period of 2 years and a month.

To calculate the amount Ken borrowed,

We need to use the formula for compound interest:

A = P * (1 + r/n)^(nt)

Where:

A is the amount after t years

P is the principal amount (the amount borrowed)

r is the annual interest rate (8% in this case)

n is the number of times the interest is compounded in a year (4 times in this case, since the interest is compounded quarterly)

t is the number of years

We know the amount after 2 years and 1 month,

So we can use that information to solve for the principal amount P.

First, we need to convert the number of years and months into a single value in terms of years:

2 years and 1 month = 2 + 1/12 = 2.0833 years

Next, we can plug in the values into the formula:

A = P * (1 + r/n)^(nt)

A = P * (1 + 0.08/4)^(4 * 2.0833)

A = P * (1.02)^(8.3333)

A = P * 1.2288

We also know that Ken is paying P2,500 every 3 months,

So we can multiply that by 4 (since there are 4 quarters in a year) to find the annual payment:

P2,500 * 4 = P10,000

And we know that A = P * 1.2288,

So we can substitute in the values we have:

P10,000 = P * 1.2288

Now we can solve for P:

P = P10,000 / 1.2288

P = 8077.84

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The line of best fit gives a general outlook on the data while the correlation is the exact points showcased to calculate or show for a data set or table.

It should be noted that between the two variables it is a positive correlation because they both increase in the same direction.  

Positive correlation is a relationship between two variables in which both variables move in tandem that is, in the same direction.

In order to find the residual one would subtract the predicted value from the measured value.

The diagram is attached.

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So x/1 + y/-2 = 1 and x/1 + y/-2 = 1 are the slope intercept forms of graphs 1, respectively.

The optimal angle in geometry is where the line's incline contacts the y-axis. a point where a line or curve's y-axis crosses it. This is demonstrated using the equation and for straight line, Y = shifting from traditional, where m denotes the slope and c the en la. The line's slope (m) as well as y-intercept (b) are highlighted in the analytic form of the equation. The slope is feet and the y-intercept is b when an equation has to have the intercept form (y=mx+b). It is also reasonable to rewrite some equations so that they appear to be slope intercepts. For example, the inclination and y-intercept are both modified to 1 if y=x is rewritten as y=1x+0.

given

The slope intercept form = x/a + y /b = 1

1) for graph 1

The intercept form is  x/1 + y/-2 = 1

2) for graph 2

The intercept form is x /-1 + y /1 = 1

So x/1 + y/-2 = 1 and x/1 + y/-2 = 1 are the slope intercept forms of graphs 1, respectively.

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Check the picture below.

so to get the EQUATion of each lines, we'll use those points you see in the picture for each, now we're only getting their "equation" only just yet, then we'll do the inequality part.

for the blue line

[tex](\stackrel{x_1}{0}~,~\stackrel{y_1}{-5})\qquad (\stackrel{x_2}{5}~,~\stackrel{y_2}{0}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{0}-\stackrel{y1}{(-5)}}}{\underset{\textit{\large run}} {\underset{x_2}{5}-\underset{x_1}{0}}} \implies \cfrac{0 +5}{5} \implies \cfrac{ 5 }{ 5 } \implies 1[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-5)}=\stackrel{m}{ 1}(x-\stackrel{x_1}{0}) \implies y +5 = 1 ( x -0) \\\\\\ y+5=x\implies \boxed{y=x-5}[/tex]

for the red line

[tex](\stackrel{x_1}{-4}~,~\stackrel{y_1}{3})\qquad (\stackrel{x_2}{6}~,~\stackrel{y_2}{-2}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-2}-\stackrel{y1}{3}}}{\underset{\textit{\large run}} {\underset{x_2}{6}-\underset{x_1}{(-4)}}} \implies \cfrac{-5}{6 +4} \implies \cfrac{ -5 }{ 10 } \implies - \cfrac{ 1 }{ 2 }[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{3}=\stackrel{m}{- \cfrac{ 1 }{ 2 }}(x-\stackrel{x_1}{(-4)}) \implies y -3 = - \cfrac{ 1 }{ 2 } ( x +4) \\\\\\ y-3=- \cfrac{ 1 }{ 2 }x-2\implies \boxed{y=- \cfrac{ 1 }{ 2 }x+1}[/tex]

now, the dashed line for the red one, means the borderline is not included so whatever "y" is, is either > or <.

the solid line for the blue line means, the borderline is included, so whatever "y" is, is ⩾ or ⩽.

now, what area do we shaded, let's deal with the red one first.

well, do usually a true/false region check, so we pick a point on either side of the line, hmmmm for simplicity let's pick the origin, (0,0), which is below the red line, that means x = 0 and y = 0

[tex]y ~~ \square- \cfrac{ 1 }{ 2 }x+1\implies 0~~ \square-\cfrac{1}{2}(0)+1\implies 0~~ \square ~~ 1\implies 0 < 1[/tex]

so the sign that will make that statement true can only possible by "<", meaning that "0 is less than 1", that means that equation is

[tex]{\Large \begin{array}{llll} y ~~ < - \cfrac{ 1 }{ 2 }x+1 \end{array}}[/tex]

now let's deal with the blue line.

same gig, we'll do a true/false region check, hmm let's pick for the sake of simplicity and slacking the same point, (0,0) which is above the blue line

[tex]y ~~ \square ~~ x-5\implies 0~~ \square ~~0-5\implies 0~~ \square ~~-5\implies 0\geqslant -5[/tex]

so only sign that makes that true is "⩾" for the blue line, because "0 is indeed greater or equal than -5", so we get

[tex]{\Large \begin{array}{llll} y \geqslant x-5 \end{array}}[/tex]

now, bear in mind that we could have pick some other point, on either side, and the issue is, to make it a true or false statement by using either inequality, for example if we end up with two values such as -9 [  ] -1, well, -9 is lesser than -1, so it can only be -9 < -1 or -9 ⩽ -1 to make it true, now if we want to make that statement false, we simply do -9 > -1 or such, and that part is "no shaded", because is false, -9 is not greater than -1.

15

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