Question 178
O conjunto solução da inequação $2x - 3 > 7$ é
(A) $\{x \in \mathbb{R} \mid x < 5\}$ (B) $\{x \in \mathbb{R} \mid x > 5\}$ (C) $\{x \in \mathbb{R} \mid x < 2\}$ (D) $\{x \in \mathbb{R} \mid x > 2\}$ (E) $\{x \in \mathbb{R} \mid x > 10\}$

O conjunto solução da inequação $2x - 5 > 3$ no conjunto dos números reais é
(A) $\{x \in \mathbb{R} \mid x < 4\}$ (B) $\{x \in \mathbb{R} \mid x > 4\}$ (C) $\{x \in \mathbb{R} \mid x < -4\}$ (D) $\{x \in \mathbb{R} \mid x > -4\}$ (E) $\{x \in \mathbb{R} \mid x = 4\}$

QUESTION 148
The solution set of the inequality $2x - 5 > 3$ is
(A) $x > 1$
(B) $x > 2$
(C) $x > 3$
(D) $x > 4$
(E) $x > 5$

In building construction, tubes of different sizes are used for water network installation. These measurements are known by their diameter, often measured in inches. Some of these tubes, with measurements in inches, are tubes of $\frac{1}{2}, \frac{3}{8}$ and $\frac{5}{4}$. Placing the values of these measurements in increasing order, we find
(A) $\frac{1}{2}, \frac{3}{8}, \frac{5}{4}$
(B) $\frac{1}{2}, \frac{5}{4}, \frac{3}{8}$
(C) $\frac{3}{8}, \frac{1}{2}, \frac{5}{4}$
(D) $\frac{3}{8}, \frac{5}{4}, \frac{1}{2}$
(E) $\frac{5}{4}, \frac{1}{2}, \frac{3}{8}$

A region of a factory must be isolated, as employees are exposed to accident risks there. This region is represented by the gray portion (quadrilateral with area S) in the figure.
So that employees are informed about the location of the isolated area, informational posters will be posted throughout the factory. To create them, a programmer will use software that allows drawing this region from a set of algebraic inequalities.
The inequalities that should be used in the said software for drawing the isolation region are
(A) $3y - x \leq 0 ; 2y - x \geq 0 ; y \leq 8 ; x \leq 9$
(B) $3y - x \leq 0 ; 2y - x \geq 0 ; y \leq 9 ; x \leq 8$
(C) $3y - x \geq 0 ; 2y - x \leq 0 ; y \leq 9 ; x \leq 8$
(D) $4y - 9x \leq 0 ; 8y - 3x \geq 0 ; y \leq 8 ; x \leq 9$
(E) $4y - 9x \leq 0 ; 8y - 3x \geq 0 ; y \leq 9 ; x \leq 8$

For an airplane to be authorized to land at an airport, the aircraft must necessarily satisfy the following safety conditions:
I. the wingspan of the aircraft (greatest distance between the tips of the airplane's wings) must be at most equal to the width of the runway;
II. the length of the aircraft must be less than 60 m;
III. the maximum load (sum of the masses of the aircraft and its cargo) cannot exceed 110 t.
Suppose that the largest runway at this airport is 0.045 km wide, and that the models of airplanes used by airlines that use this airport are given in the table.

Model	Dimensions (length $\times$ wingspan)	Maximum load
A	$44{,}57 \mathrm{~m} \times 34{,}10 \mathrm{~m}$	110000 kg
B	$44{,}00 \mathrm{~m} \times 34{,}00 \mathrm{~m}$	95000 kg
C	$44{,}50 \mathrm{~m} \times 39{,}50 \mathrm{~m}$	121000 kg
D	$61{,}50 \mathrm{~m} \times 34{,}33 \mathrm{~m}$	79010 kg
E	$44{,}00 \mathrm{~m} \times 34{,}00 \mathrm{~m}$	120000 kg

The only aircraft able to land at this airport, according to safety regulations, are those of models
(A) A and C.
(B) A and B.
(C) B and D.
(D) B and E.
(E) C and E.

On a map with scale 1 : 250 000, the distance between cities A and B is 13 cm. On another map, with scale 1 : 300 000, the distance between cities A and C is 10 cm. On a third map, with scale 1 : 500 000, the distance between cities A and D is 9 cm. The actual distances between city A and cities B, C, and D are, respectively, equal to $X$, $Y$, and $Z$ (in the same unit of length).
The distances $X$, $Y$, and $Z$, in increasing order, are given in
(A) $X, Y, Z$.
(B) $Y, X, Z$.
(C) $Y, Z, X$.
(D) $Z, X, Y$.
(E) $Z, Y, X$.

The owner of a restaurant wants to buy a rectangular glass top for the base of a table, as illustrated in the figure.
It is known that the base of the table, considering the outer edge, has the shape of a rectangle, whose sides measure $AC = 105 \mathrm{~cm}$ and $AB = 120 \mathrm{~cm}$.
At the store where the top will be purchased, there are five types of top options with different dimensions, all with the same thickness, namely:
Type 1: $110 \mathrm{~cm} \times 125 \mathrm{~cm}$
Type 2: $115 \mathrm{~cm} \times 125 \mathrm{~cm}$
Type 3: $115 \mathrm{~cm} \times 130 \mathrm{~cm}$
Type 4: $120 \mathrm{~cm} \times 130 \mathrm{~cm}$
Type 5: $120 \mathrm{~cm} \times 135 \mathrm{~cm}$
The owner evaluates, for user convenience, that one should choose the top with the smallest possible area that satisfies the condition: when placing the top on the base, on each side of the outer edge of the table base, there should be a remaining region, corresponding to a glass frame, limited by a minimum of 4 cm and maximum of 8 cm outside the table base, on each side.
According to the previous conditions, which type of glass top did the owner evaluate should be chosen?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

A rectangular plot of land with sides whose measurements, in meters, are $x$ and $y$ will be fenced for the construction of an amusement park. One side of the plot is located on the banks of a river. Observe the figure.
To fence the entire plot, the owner will spend R\$ 7500.00. The fence material costs R\$ 4.00 per meter for the sides of the plot parallel to the river, and R\$ 2.00 per meter for the other sides.
Under these conditions, the dimensions of the plot and the total cost of the material can be related by the equation
(A) $4(2x + y) = 7500$
(B) $4(x + 2y) = 7500$
(C) $2(x + y) = 7500$
(D) $2(4x + y) = 7500$
(E) $2(2x + y) = 7500$

Until November 2011, there was no specific law that punished fraud in public competitions. This made it difficult to frame fraudsters in any specific article of the Penal Code, causing them to escape Justice more easily. However, with the enactment of Law 12.550/11, it is considered a crime to improperly use or disclose confidential content of a public competition, with a penalty of imprisonment of 12 to 48 months (1 to 4 years). If this crime is committed by a public official, the penalty will increase by $\frac{1}{3}$.
If a public official is convicted of defrauding a public competition, his imprisonment sentence may vary from
(A) 4 to 16 months.
(B) 16 to 52 months.
(C) 16 to 64 months.
(D) 24 to 60 months.
(E) 28 to 64 months.

The manager of a parking lot near a large airport knows that a passenger who uses his car for home-airport-home trips spends about $\mathrm{R}\$ 10.00$ on fuel for this trip. He also knows that a passenger who does not use his car for home-airport-home trips spends about $\mathrm{R}\$ 80.00$ on transportation.
Suppose that passengers who use their own vehicles leave their cars in this parking lot for a period of two days.
To make the use of the parking lot attractive to these passengers, the value, in reais, charged per day of parking should be, at most,
(A) 35.00.
(B) 40.00.
(C) 45.00.
(D) 70.00.
(E) 90.00.

The solution set of the inequality $2x - 3 > 7$ is:
(A) $x > 2$
(B) $x > 3$
(C) $x > 4$
(D) $x > 5$
(E) $x > 6$

System of inequalities $$\left\{ \begin{array} { l } \frac { x + 2 } { x ^ { 2 } - 4 x + 3 } \geqq 0 \\ \frac { 9 } { x - 8 } \leqq - 1 \end{array} \right.$$ What is the number of integers $x$ that satisfy the system? [2 points]
(1) 10
(2) 9
(3) 8
(4) 7
(5) 6

For two natural numbers $a , b ( a < b )$, the fractional inequality
$$\frac { x } { x - a } + \frac { x } { x - b } \leqq 0$$
is satisfied by exactly 2 integers $x$. What is the number of ordered pairs $( a , b )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

System of inequalities $$\left\{ \begin{array} { l } x ( x - 4 ) ( x - 5 ) \geqq 0 \\ \frac { x - 3 } { x ^ { 2 } - 3 x + 2 } \leqq 0 \end{array} \right.$$ What is the number of integers $x$ that satisfy the system? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Two quadratic expressions $f ( x ) , g ( x )$ with leading coefficient 1 have greatest common divisor $x + 3$ and least common multiple $x ( x + 3 ) ( x - 4 )$. How many integers $x$ satisfy the fractional inequality $\frac { 1 } { f ( x ) } + \frac { 1 } { g ( x ) } \leqq 0$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For the fractional inequality in $x$ $$1 + \frac { k } { x - k } \leqq \frac { 1 } { x - 1 }$$ Find the value of the natural number $k$ such that the number of integers $x$ satisfying the inequality is 3. [3 points]

For two sets
$$A = \left\{ x \left\lvert \, \frac { ( x - 2 ) ^ { 2 } } { x - 4 } \leq 0 \right. \right\} , \quad B = \left\{ x \mid x ^ { 2 } - 8 x + a \leq 0 \right\}$$
When $A \cup B = \{ x \mid x \leq 5 \}$, what is the value of the constant $a$? [3 points]
(1) 7
(2) 10
(3) 12
(4) 15
(5) 16

The product of all real roots of the irrational equation $x ^ { 2 } - 2 x + 2 \sqrt { x ^ { 2 } - 2 x } = 8$ is? [3 points]
(1) - 5
(2) - 4
(3) - 3
(4) - 2
(5) - 1

Starting from point A, one travels to point B which is 6 km away, and then returns to point A along the same route. For the first 1 km, one walks at a constant speed, and for the remaining 5 km, one travels at twice the initial walking speed. On the return trip, one travels at a speed 2 km/h faster than the initial walking speed. When the total time for the round trip is at most 2 hours 30 minutes, what is the minimum value of the initial walking speed? (Given that the unit of speed is km/h.) [3 points]
(1) $\frac { 12 } { 5 }$
(2) $\frac { 13 } { 5 }$
(3) $\frac { 14 } { 5 }$
(4) 3
(5) $\frac { 16 } { 5 }$

For the logarithmic inequality in $x$ $$\log _ { 5 } ( x - 1 ) \leq \log _ { 5 } \left( \frac { 1 } { 2 } x + k \right)$$ when the number of all integers $x$ satisfying this inequality is 3, what is the value of the natural number $k$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two conditions on the real number $x$: $$\begin{aligned} & p : | x - 1 | \leq 3 , \\ & q : | x | \leq a \end{aligned}$$ What is the minimum value of the natural number $a$ such that $p$ is a sufficient condition for $q$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two conditions on real number $x$: $$\begin{aligned} & p : ( x - 1 ) ( x - 4 ) = 0 , \\ & q : 1 < 2 x \leq a \end{aligned}$$ Find the minimum value of the natural number $a$ such that $p$ is a sufficient condition for $q$. [3 points]
(1) 4
(2) 5
(3) 6
(4) 7
(5) 8

For two conditions $p$ and $q$ on real numbers $x$: $$\begin{aligned} & p : x ^ { 2 } - 4 x + 3 > 0 , \\ & q : x \leq a \end{aligned}$$ What is the minimum value of the real number $a$ such that $\sim p$ is a sufficient condition for $q$? [3 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

Find the maximum value of the real number $k$ such that the graphs of $y = \sqrt { x + 3 }$ and $y = \sqrt { 1 - x } + k$ intersect. [4 points]