Answer:
B
Step-by-step explanation:
Add exponents when the bases are multiplied.
a^2 * a ^5
a^(2 + 5)
a^7
Answer:
B. 7
Step-by-step explanation:
In the triangle below, what is the measure of Angle z
Answer:65
Step-by-step explanation:
The triangle is isosceles triangle and the base angles must be the same
The measure of angle z is 65°.
Given that, in ΔXYZ XZ=XY=7 cm.
We need to find the value of ∠Z.
What is the isosceles triangle?In an isosceles triangle with two equal sides. The angles opposite the equal sides are also equal.
Since XZ=XY=7 cm.
Here, ∠Y=∠Z=65°
Therefore, the measure of angle z is 65°.
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What is the simplified form of sqrt 10,000
Answer:
100
Step-by-step explanation:
10,000 is merely 100 x 100 = 100²
so √10,000 = √100² = 100
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
A number cube is rolled 450 times. The number 3 comes up 67 times.
a. What is the theoretical probability of rolling a 3? Write your answer as a fraction in simplest form.
b. What is the experimental probability of rolling a 3? Write your answer as a fraction in simplest form.
Answer:
Theoretical P(3) = 1/6
Experimental P(3) = 67/450
Step-by-step explanation:
A die has 6 possibilities that can come up {1,2,3,4,5,6} They are all equally likely, so each has a probability of 1/6
a P(getting a 3) = 1/6
We rolled a 3 67 times out of 450. This is our experimental probability
b P (getting a 3) = 67/450
This does not simplify
As your math teacher, let's solve this problem together step by step.
a. Theoretical Probability of Rolling a 3:
A standard number cube, often referred to as a die, has six faces numbered 1 through 6. Since all faces are usually designed to be equally probable, the theoretical probability of rolling a three (or any specific number) is the ratio of the number of successful outcomes (rolling a 3) to the total number of possible outcomes (any number from 1 to 6).
So,
Theoretical Probability = Number of successful outcomes / Total number of possible outcomes
= 1 / 6
This is already in the simplest form since 1 and 6 have no common divisors other than 1.
b. Experimental Probability of Rolling a 3:
The experimental probability, in contrast to the theoretical probability, is determined based on actual trials or experiments. In this case, the cube was rolled 450 times, and the number 3 appeared 67 times.
So,
Experimental Probability = Number of times a 3 was rolled / Total number of rolls
= 67 / 450
Now, to simplify the fraction, you would find the greatest common divisor (GCD) of 67 and 450 and divide both numerator and denominator by that GCD.
Looking at the numbers, they do not immediately suggest a common factor, and since 67 is a prime number, we can reasonably conclude that the fraction is already in its simplest form. Therefore, the experimental probability is:
= 67 / 450
So there you have it, answers to both questions in their simplest form:
a. The theoretical probability of rolling a 3 on a number cube is 1/6.
b. The experimental probability of rolling a 3 given the cube is rolled 450 times and 3 comes up 67 times is 67/450.
A swim teacher sells lesson packages. The best deal has the highest ratio of lessons to total cost.
Swim Lesson Packages
Number of Lessons Total Cost
$10
$40
10 1 $80
15
$80
Which package is the best deal?
l lesson for $10
5 lessons for $40
10 lessons for $80
15 lessons for $80
Mark this and return
Save and Exit
Answer:
the best package is 15 lessons for $80
Answer:
Which package is the best deal? Option D
(D) 15 lessons for $80 100% Answer
Step-by-step explanation:
Solve the following word problems using your knowledge
9 word problems using your knowledge of fractions. Be sure to express your answers
in simplest form.
Mike had 9 1/2 cups of Gatorade in a bottle. After he darom 2 3/4 cups of Gatorade, how much was left in the bottle?
can you please help me and I will get you brainliest.
Answer:
27/4 or 6 3/4
Step-by-step explanation:
( 19 x 4 ) - ( 11 x 2 )
------------------------------
2 x 4
=54/8
54/2
-------------
8/2
A cylinder has a volume of 288x cubic meters and a height of 9 meters. What is the area of the base?
Answer: area of the base = 32 m²
Step-by-step explanation:
Cylinder volume is the product of area of the base by height.
Then, area of the base = cylinder volume/height = 288m³/9m = 32 m²
Answer: area of the base = 32 m²
[tex]\textit{\textbf{Spymore}}[/tex]
5x sqaure plus 25 divided by 9
Answer:
5 • (9x2 + 5)
———————
9
Step-by-step explanation:
Step 1 :
25
Simplify ——
9
Equation at the end of step 1 :
25
(5 • (x2)) + ——
9
Step 2 :
Equation at the end of step 2 :
25
5x2 + ——
9
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 Adding a fraction to a whole
Rewrite the whole as a fraction using 9 as the denominator :
5x2 5x2 • 9
5x2 = ——— = ———————
1 9
Hope this helps! Please mark brainliest!
Which of the following pair(s) of circles have / as a common external tangent? Select all that apply.
A and B
A and C
B and C
Answer with explanation:
A tangent to a circle is a Line which touches a circle at a single point.
There can be Infinite number of tangents to a circle which are external only.
In the given figure:
⇒ There are three circles having Centers A, B and C.Circle having Center B lies inside the circle having center C.
⇒The three circles intersect at point T.
⇒The common tangent of these three circles passes through Point T.
⇒Circles having Center B and Center C , have a common external Tangent.
Option A,B and C:
→A and B
→A and C
→B and C
Tangent is the line that intersects the circle at a single point. Thus, Option A, B, and C all apply.
Three different circles are given in the question that is A, B, and C.
We have to determine the pair of circles that have a common external tangent.
Therefore, take circles A and B from the given figure.
Line " L " is the common tangent to both circles. It is clear from the figure that it is an external tangent to both circles.
Hence, Option A applies.
Now, take circles A and C from the given figure.
Line " L " is the common tangent to both circles. It is clear from the figure that it is an external tangent to both circles.
Hence, Option B applies.
Further, take the circles B and C from the given figure.
Line " L " is the common tangent to both circles. It is clear from the figure that it is an external tangent to both circles.
Hence, Option C applies.
Thus, Option A, B, and C all apply.
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This time make a simple coaster that bumps the axis at x=500. Remember to make sure that the track rises before it falls
Answer:
y=-ax(x-500)^2(x-1000)
Step-by-step explanation:
Calculate the approximate mean absolute deviation of the given data set.
5, 7, 8, 10, 11, 17, 19, 24
8
12.625
5.53
10.5
Answer:
12.625
mean=total sum value /total data set
The mean of given data set is: 12.625.
The correct option is (B)
What is mean?
Mean is the average of the given numbers and is calculated by dividing the sum of given numbers by the total number of numbers.
Mean = (Sum of all the observations/Total number of observations)
Given data: 5,7,8,10,11,17,19,24.
Mean=
sum of observation/ Total number of observation
=5+7+8+10+11+17+19+24/8
=101/8
=12.625.
Hence, the mean is 12.625.
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A line is defined by the equation y = 2/3 x - 6 The line passes through a point whose y-coordinate is 0. What is the x-coordinate of this point?
Answer:
x = 9
Step-by-step explanation:
Use the equation of the line, and let y = 0. Then solve for x.
y = 2/3 x - 6
Let y = 0:
0 = 2/3 x - 6
Add 6 to both sides.
6 = 2/3 x
Multiply both sides by 3/2.
3/2 * 6 = x
x = 9
What is the range of the function f(x) = 1/2 square root of x ? Anyone
Answer:
The range is f(x) ≥ 0
Step-by-step explanation:
The range of the function is defined as the set of values of the dependent variable for which the function is defined.
Here the function f(x) = 1/2 √x is defined for all values of x which are greater than or equal to zero or we can say all non-negative real numbers.
So, the range is f(x) ≥ 0
Final answer:
The range of the function f(x) = ½√x is all real numbers greater than or equal to 0, which is written in interval notation as [0, ∞).
Explanation:
The range of a function is the set of all possible output values it can produce. For the function f(x) = ½√x, the domain (input values) must be non-negative because we cannot take the square root of a negative number in the real number system. Since the smallest non-negative number is 0 and the value of the function at x=0 is f(0) = ½√0 = 0, the function starts at 0. The square root function increases as its input increases; thus, for any positive value of x, you get a positive value for f(x). Moreover, as x approaches infinity, the output of the function also grows without any upper bound, although it does so slower than the increase in x.
Therefore, the range of the function f(x) = ½√x is all real numbers greater than or equal to 0, which can be written in interval notation as [0, ∞).
Simplify the following in the form of a +b √c
1- √5/1+√5
Answer:
see explanation
Step-by-step explanation:
Given
[tex]\frac{1-\sqrt{5} }{1+\sqrt{5} }[/tex]
Multiply the numerator and denominator by the conjugate of the denominator
The conjugate of 1 + [tex]\sqrt{5}[/tex] is 1 - [tex]\sqrt{5}[/tex], hence
[tex]\frac{(1-\sqrt{5})(1-\sqrt{5}) }{(1+\sqrt{5})(1-\sqrt{5}) }[/tex]
Expand numerator/ denominator
= [tex]\frac{1-2\sqrt{5}+5 }{1-5}[/tex]
= [tex]\frac{6-2\sqrt{5} }{-4}[/tex]
= [tex]\frac{6}{-4}[/tex] + [tex]\frac{-2\sqrt{5} }{-4}[/tex]
= - [tex]\frac{3}{2}[/tex] + [tex]\frac{1}{2}[/tex] [tex]\sqrt{5}[/tex]
Answer:
-3/2 + (1/2)*sqrt(5)
Note: a=-3/2 , b=1/2 , c=5
Step-by-step explanation:
I think this is meant to be written as (1-sqrt(5))/(1+sqrt(5)).
First step: Multiply top and bottom by the conjugate of the bottom which is 1-sqrt(5).
When you multiply conjugates, you do have to do the whole foil thing... just do first and last because the others will cancel.
So what I'm saying is when multiply (1+sqrt(5))(1-sqrt(5)) you will get 1-5=-4.
Second step: Multiply top out... you have to do the whole foil here because you aren't multiplying conjugates. So (1-sqrt(5))(1-sqrt(5))=1-sqrt(5)-sqrt(5)+5=
1-2sqrt(5)+5=6-2sqrt(5).
Third step: Second step/first step =(6-2sqrt(5))/-4
Fourth step: Separate fraction 6/-4 - 2sqrt(5)/-4
Fifth step: Simplify each fraction: -3/2 + (1/2)*sqrt(5)
Sixth step: If you compare the form you want it in to the form I wrote my answer in, then you should see that a=-3/2 , b=1/2 , c=5
what two integers is √42 between? explain
[tex]\sqrt{36}=6[/tex] and [tex]\sqrt{49}=7[/tex], so it stands to reason that [tex]\sqrt{42}[/tex] lies somewhere between 6 and 7.
Will give brainliest!! Part of me thinks the answer is D but I just want to double check.
Answer: C) 6.0025
Step-by-step explanation:
The equation for the area of a circle requires the radius, you put in the full diameter. Just divide 4.9 by 2, which gives you 2.45. 2.45 squared is equal to 6.0025 and then slap your pi on to the end of it.
Answer:
[tex]6.0025\pi \: {m}^{2} [/tex]
Step-by-step explanation:
The formula for the area of a circle:
[tex]\pi \times {r}^{2} [/tex]
We are given the diameter of the circle, which is two times its radius. To find the radius, we divide the diameter by 2.
4.9 ÷ 2 = 2.45 m^2.
Now we need to square the radius:
[tex] {2.45}^{2} = 6.0025[/tex]
So, the area in terms of pi would be
[tex]\pi \ \times {6.0025}^{2} \: {m}^{2} [/tex]
Or
[tex]6.0025 \times \pi \: {m}^{2} [/tex]
Hope this helps!
How do you do right Triangle
Answer:
right triangles should have one right angle (90 degrees) and it should liik like an L, and then draw a line connecting the gap (the hypotenuse)
Step-by-step explanation:
Answer:
make a 90 degree angle and connect the two points to finish the shape of the triangle.
Step-by-step explanation:
HINT: the corner of a piece of paper is a 90 degree angle
Which function has only one x-intercept at (-6, 0)?
Of(x) = x(x - 6)
f(x) = (x - 6)(x – 6)
f(x) = (x + 6)(x - 6)
f(x) = (x + 1)(x + 6)
Answer:
The function f(x) = (x - 6)(x - 6) has only one x-intercept. But at (6, 0) not at (-6, 0).Step-by-step explanation:
The intercept form of a quadratic equation (parabola):
[tex]y=a(x-p)(x-q)[/tex]
p, q - x-intercepts
Therefore
The function f(x) = x(x - 6) = (x - 0)(x - 6) has two x-intercepts at (0, 0) and (6, 0)
The function f(x) = (x - 6)(x - 6) has only one x-intercept at (6, 0)
The function f(x) = (x + 6)(x - 6) = (x - (-6))(x - 6)
has two x-intercept at (-6, 0) and (6, 0)
The function f(x) = (x + 1)(x + 6) = (x - (-1))(x - (-6))
has two x-intercepts at (-1, 0) and (-6, 0).
Answer:
the answer is d on edge
Step-by-step explanation:
Which expression is the factorization of x2 + 10x + 21?
You must remember that a polynomial is written like so...
ax^2 + bx + c
In this case...
a = 1
b = 10
c = 21
To factor you must find two numbers who both add up to b (10) AND multiply to c (21)
3 + 7 = 10
3 * 7 = 21
so...
(x + 3 )(x + 7)
Hope this helped!
~Just a girl in love with Shawn Mendes
Answer: 12x + 21 or C. F(x) = (x - 2)(x + 2)
Step-by-step explanation:
x2 + 10x + 21
2x + 10x + 21
Rearrange to put the coefficient in front/move it in front of the term.
Ex. (× · 4 ⇒ 4×)
(×( 7 ) ⇒ 7×)
Identify and group like terms.
(2x + 10x) + 21
Group the coefficients together.
(2+ 10)x + 21
Combine 2 + 10 to get 12.
12x + 21
= 12x + 21
You're Welcome! :))
solve the system .....................
Answer:
The correct answer is first option
(2, 3)
Step-by-step explanation:
It is given that,
3y = -x + 11 and
x + 4y = 14
To find the solutions
The given equation can be written as,
x + 3y = 11 --------(1)
x + 4y = 14 -------(2)
Subtract (1) from (2) we get
y = 3
Substitute vale of y in eq(1)
x + 3*3 = 11
x = 11 - 9 = 2
Therefore x = 2 and y = 3
The correct answer is first option
Answer:
first option
(2, 3)
Step-by-step explanation:
We have the following system of linear equations
[tex]\left \{{{3y=-x+11} \atop {x+4y=14}} \right.[/tex]
This is the same as
[tex]\left \{{{x + 3y=11} \atop {x+4y=14}} \right.[/tex]
To solve the system multiply the first equation by -1 and then add it to the second equation.
[tex]-1*(x+3y)=11*(-1)[/tex]
[tex]-x -3y=-11\\x+4y=14[/tex]
-----------------
[tex]y = 14-11[/tex]
[tex]y = 3[/tex]
substitute [tex]y = 3[/tex] in any of the two equations and then solve for x
[tex]x+4(3)=14\\x+12=14\\x =14-12\\x = 2\\[/tex]
[tex]x =2[/tex]
The answer is the first option
(2, 3)
Would the answer for 36 be $9.88 or $9.75?
total bill before any discounts $13.
she is a student, so she gets 20% off.
she brought an item of clothing, so she gets 5% off.
so she's really getting 20% + 5% off, namely 25% off her bill.
[tex]\bf \begin{array}{|c|ll} \cline{1-1} \textit{a\% of b}\\ \cline{1-1} \\ \left( \cfrac{a}{100} \right)\cdot b \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{25\% of 13}}{\left( \cfrac{25}{100} \right)13\implies 3.25}~\hfill \stackrel{\textit{bill with all discounts}}{13-3.25\implies 9.75}[/tex]
find the nature of the quadratic equation 2x^2+5x+8=0
[tex]
2x^2+5x+8=0 \\
x=\dfrac{-5+\vee-\sqrt{5^2-2\cdot2\cdot8}}{2\cdot2} \\
x=\dfrac{-5+\vee-\sqrt{25-64}}{4} \\
x=\dfrac{-5+\vee-\sqrt{-39}}{4}\Longrightarrow x\notin\mathbb{R} \\
x\notin\mathbb{R}\Longrightarrow x\in\mathbb{C}
[/tex]
So no real solutions for x. But two complex solutions for x.
[tex]
x_1=\boxed{\dfrac{-5-39i}{4}} \\ \\
x_2=\boxed{\dfrac{-5+39i}{4}}
[/tex]
Hope this helps.
r3t40
which of the following is a three dimensional solid has a circle as its cross section?
cone
pyramid
prism
cone
Answer:
A cone is a three dimensional solid with a circle at its cross section
Answer:
The answer is cone.
Step-by-step explanation:
A cone is a three dimensional figure, whose cross section parallel to its base, gives a circle because the base of the cone is a circle.
A cross section is defined as the slicing of a three dimensional figure to get a two dimensional figure.
Here, cone is a three dimensional figure and circle is a two dimensional figure.
So, the answer is cone.
Find the value of Q in the following system so that the solution to the system is (2,3)
x-2y=-4
3x+5y=Q
Answer:
Q = 21Step-by-step explanation:
[tex]\left\{\begin{array}{ccc}x-2y=-4&(1)\\3x+5y=Q&(2)\end{array}\right\\\\\text{From (2, 3), put x = 2 and y = 3 to (2):}\\\\Q=3(2)+5(3)=6+15=21[/tex]
Which expression is equivalent to the expression below?
Answer:
A
Step-by-step explanation:
Given
[tex]\sqrt{-108}[/tex] - [tex]\sqrt{-3}[/tex]
= [tex]\sqrt{36(3)(-1)}[/tex] - [tex]\sqrt{3(-1)}[/tex]
= 6[tex]\sqrt{3}[/tex] i - [tex]\sqrt{3}[/tex] i
= 5[tex]\sqrt{3}[/tex] i
Answer:
Step-by-step explanation:
√(-108) - √(-3)
= √(-3*36) - √(-3)
= √[(-3)*36] - √(-3)
= √(-3)*(√36 - 1)
= √[(-1)*(3)]*(6-1)
= i*√(3)*(5)
= 5√3i
The domain for f(x) and g(x) is the set of all real numbers.
Let f(x)=3x+5 and g(x)=x^2 .
Find g(x)-f(x)
A.) 3x^2-5
B.) x^3-5
C.) 3x^3-5x^2
D.) x^2-3x-5
Please an explanation too:) Thank you!
Answer:
D.) x^2-3x-5
Step-by-step explanation:
f(x)=3x+5
g(x)=x^2 .
Find g(x)-f(x)
g(x) - f(x)= x^2 -(3x+5)
Distribute the minus sign
x^2 -3x-5
Answer:
x² - 3x - 5 ⇒ answer D
Step-by-step explanation:
* Lets explain how to solve the problem
- There are two functions f(x) ang g(x)
- f(x) = 3x + 5 ⇒ it is a linear function
- g(x) = x² ⇒ it is a quadratic function
- Both functions have a domain the set of real numbers
- We want to subtract f(x) from g(x)
* Lets solve the problem
∵ g(x) = x²
∵ f(x) = 3x + 5
∵ f(x) will subtracted from g(x)
∴ g(x) - f(x) = (g - f)(x)
- Lets make the subtraction
∴ (g - f)(x) = x² - (3x + 5)
- Open the bracket by multiply the negative sign by the two terms
of the bracket
∵ -(3x) = - 3x
∵ -(5) = - 5
∴ (g - f)(x) = x² - 3x - 5
∴ g(x) - f(x) = x² - 3x - 5
If $740 is invested at an interest rate of 11% per year and is compounded continuously, how much will the investment be worth in 7 years?
Use the continuous compound interest formula A = Pert.
Answer:
[tex]\$1,598.23[/tex]
Step-by-step explanation:
we know that
The formula to calculate continuously compounded interest is equal to
[tex]A=P(e)^{rt}[/tex]
where
A is the Final Investment Value
P is the Principal amount of money to be invested
r is the rate of interest in decimal
t is Number of Time Periods
e is the mathematical constant number
we have
[tex]t=7\ years\\ P=\$740\\ r=0.11[/tex]
substitute in the formula above
[tex]A=\$740(e)^{0.11*7}[/tex]
[tex]A=\$1,598.23[/tex]
Find. F(g(d))
F(d)=d-15
G(d)=0.8d
Since f(d) and g(d) represent two types of discounts that you will receive off of the price, the composite function is f(g(d)) = 0.8d - 15.
In Mathematics and Geometry, a function composition is an operation (∘) that combines two functions f(x) and g(x), in order to produce a composite function h(x) = (g∘f)(x), such that h(x) = g.
In this exercise, we would determine the corresponding output values for the composite functions by using the substitution method as follows;
f(g(d)) = 0.8d - 15
In this context, we can reasonably infer and logically deduce that the composite function is f(g(d)) = 0.8d - 15.
Complete Question:
You go to a store to buy sneakers f(d) and g(d) represent two types of discounts that you will receive off of the price of the sneakers that you want.
Find f(g(d))
f(d)=d-15
g(d)=0.8d
What’s the slope 3y=15-6x
Answer:
-2
Step-by-step explanation:
Isolate the variable y. Note the equal sign, what you do to one side, you do to the other. Divide 3 from both sides:
(3y)/3 = (15 - 6x)/3
y = (15 - 6x)/3
y = 5 - 2x
Note the equation:
y = mx + b
m = slope
b = y-intercept
x & y = the point (x , y)
Note that the slope is directly next to x. -2 is your answer.
~
For this case we have by definition, that the equation of a line in the slope-intersection form is given by:
[tex]y = mx + b[/tex]
Where:
m: It's the slope
b: It is the cutoff point with the y axis
We have the following equation:
[tex]3y = 15-6x\\y = \frac {15} {3} - \frac {6x} {3}\\y = 5-2x\\y = -2x + 5[/tex]
So we have to:
[tex]m = -2[/tex]
Answer:
The slope is -2
what is the equation of a line that goes through the point (0,5/6) and has a slope of 1
The slope is given as 1.
The Y-intercept is where the line crosses the Y axis when X is 0. This is also given by the point (0,5/6). X is 0 and Y is 5/6, so the Y-intercept is 5/6.
The equation of a line is given as y = mx +b where m is the slope and b is the y-intercept.
The equation is: y = x +5/6
Answer:
y = x+5/6
Step-by-step explanation:
The slope intercept form of the equation of a line is
y = mx +b
We are given the slope of 1 and the y intercept of 5/6 (The y intercept is when x=0)
y = 1x+5/6
y = x+5/6
which of the following points are solutions to the system of inequalities shown below?
check all that apply
y>6x+7
y<6x+9
answers:
a. (3,26)
b. (-4,32)
c. (3,25)
d. (4,33)
Answer:
see below
Step-by-step explanation:
y > 6x + 7
y < 6x + 9
check one by one
A. (3, 26)
26 > 6(3) + 7→ 26 > 25 yes
26 < 6(3) + 9→ 26 < 27 yes
B. (-4, 32)
32 > 6(-4) + 7→ 32 > -17 yes
32 < 6(-4) + 9→ 32 < -15 no
C. (3, 25)
25 > 6(3) + 7→ 25 > 25 no
25 < 6(3) + 9→ 25 < 27 yes
DB. (4, 33)
33 > 6(4) + 7→ 33 > 31 yes
33 < 6(4) + 9→ 33 < 33 no
Final answer:
Points a (3, 26) and b (-4, 32) are solutions to the system of inequalities y>6x+7 and y<6x+9. Points c (3, 25) and d (4, 33) do not satisfy both inequalities, so they are not solutions.
Explanation:
To determine which points are solutions to the given system of inequalities, y > 6x + 7 and y < 6x + 9, we need to check if they satisfy both inequalities:
Point a (3, 26): 26 > 6(3) + 7 and 26 < 6(3) + 9, which simplifies to 26 > 25 and 26 < 27. Both are true, so point a is a solution.
Point b (-4, 32): 32 > 6(-4) + 7 and 32 < 6(-4) + 9, which simplifies to 32 > -17 and 32 < -15. Both are true, so point b is also a solution.
Point c (3, 25): 25 > 6(3) + 7 and 25 < 6(3) + 9, which simplifies to 25 > 25 and 25 < 27. The first inequality is not true (it is equal, not greater), so point c is not a solution.
Point d (4, 33): 33 > 6(4) + 7 and 33 < 6(4) + 9, which simplifies to 33 > 31 and 33 < 33. Similar to point c, the second inequality is not true (it is equal, not less), so point d is not a solution.
Points a and b satisfy both inequalities and are correct solutions to the system.