Digital SAT Math Practice Questions – Medium : Linear inequalities in one or two variables

SAT MAth Practice questions – all topics

• Algebra Weightage: 35%  Questions: 13-15
• Linear equations in one variable
• Linear equations in two variables
• Linear functions
• Systems of two linear equations in two variables
• Linear inequalities in one or two variables

SAT MAth and English  – full syllabus practice tests

[calc]  Question Medium

One of the two linear equations in a system is $$3 x+4 y=8$$. The system has exactly one solution.
Which of the following could be the other equation in the system?
A) $$\frac{3}{2} x+2 y=4$$
B) $$3 x+4 y=4$$
C) $$4 x+3 y=8$$
D) $$6 x+8 y=16$$

Ans: C

To determine which equation would allow the system to have exactly one solution, the slopes of the two lines must be different. This ensures that the lines intersect at exactly one point.

The given equation is:
$3x + 4y = 8$

First, we rewrite this equation in slope-intercept form ($$y = mx + b$$) to find its slope:

$3x + 4y = 8$
$4y = -3x + 8$
$y = -\frac{3}{4}x + 2$

The slope of the given equation is $$-\frac{3}{4}$$.

Now, let’s check the slopes of the other options:

Option A: $$\frac{3}{2}x + 2y = 4$$
$\frac{3}{2}x + 2y = 4$
$2y = -\frac{3}{2}x + 4$
$y = -\frac{3}{4}x + 2$

The slope here is also $$-\frac{3}{4}$$, the same as the given equation. This means the lines are parallel and would either be identical (infinitely many solutions) or parallel (no solutions), not intersecting at one point.

Option B: $$3x + 4y = 4$$
This is the same equation as the given one but with a different constant term, which makes it a parallel line. Therefore, the lines are parallel and will not intersect at one point.

Option C: $$4x + 3y = 8$$
$4x + 3y = 8$
$3y = -4x + 8$
$y = -\frac{4}{3}x + \frac{8}{3}$

The slope here is $$-\frac{4}{3}$$, which is different from $$-\frac{3}{4}$$. This means the lines have different slopes and will intersect at exactly one point.

Option D: $$6x + 8y = 16$$
This is a multiple of the given equation:
$6x + 8y = 16$
$\frac{6x + 8y}{2} = \frac{16}{2}$
$3x + 4y = 8$

This is the same equation as the given one, just scaled by a factor of 2. Therefore, they are identical and would either have infinitely many solutions or none if they were parallel lines, not intersecting at one point.

$\boxed{C}$

[Calc]  Questions   medium

$h(x)=3 x+3$

Which inequality represents all values of $$x$$ for which the graph of $$y=h(x)$$ in the $$x y$$-plane is above the $$x$$ axis?
A. $$x<3$$
B. $$x<-1$$
C. $$x>-1$$
D. $$x>3$$

Ans:C

The function $$h(x) = 3x + 3$$ represents a linear function with a slope of $$3$$ and a $$y$$-intercept of $$3$$. The graph of $$h(x)$$ will be above the $$x$$-axis for all values of $$x$$ greater than the $$x$$-intercept, which can be found by setting $$h(x) = 0$$:

$3x + 3 = 0$
$3x = -3$
$x = -1$

Therefore, all values of $$x$$ greater than $$-1$$ will make the graph of $$h(x)$$ above the $$x$$-axis. So, the correct inequality is $$x > -1$$.

[No calc]  Question    medium

$$y\leq =2x+3$$
$$y\geq 0.5x-6$$

In which graph does the shaded region represent all solutions to the given system of inequalities?

Ans: B

Given inequalities:
1. $$y \leq 2 x+3$$
2. $$y \geq 0.5 x-6$$

1. $$y \leq 2 x+3$$ :

•  The boundary line is $$y=2 x+3$$.
• Since it is $$y \leq$$, the region below this line will be shaded.

2. $$y \geq 0.5 x-6$$ :

• The boundary line is $$y=0.5 x-6$$.
• Since it is $$y \geq$$, the region above this line will be shaded.

The correct solution will be the intersection of these two shaded regions.

Graph B:

• The region below the line $$y=2 x+3$$ is shaded. (Correct)
• The region above the line $$y=0.5 x-6$$ is shaded. (Correct)
• The intersection of these two regions is correctly shaded. (Correct)

[Calc]  Question   Medium

Sanjay works as a teacher’s assistant for $$\ 20$$ per hour and tutors privately for $$\ 25$$ per hour. Last week, he made at least $$\ 100$$ working $$x$$ hours as a teacher’s assistant and $$y$$ hours as a private tutor. Which of the following inequalities models this situation?
A) $$4 x+5 y \geq 25$$
B) $$4 x+5 y \geq 20$$
C) $$5 x+4 y \geq 25$$
D) $$5 x+4 y \geq 20$$

Ans:B

Sanjay earns $$\20$$ per hour as a teacher’s assistant and $$\25$$ per hour as a private tutor. He worked $$x$$ hours as a teacher’s assistant and $$y$$ hours as a private tutor and made at least $$\100$$.

To model this situation, we set up the following inequality:

total earnings from working $$x$$ hours as a teacher’s assistant:
$20x$

total earnings from working $$y$$ hours as a private tutor:
$25y$

3. The total earnings must be at least $$\100$$:
$20x + 25y \geq 100$

Simplify this inequality by dividing everything by 5:
$4x + 5y \geq 20$

[Calc]  Question  medium

According to a 2008 study, there were five known subspecies of tigers, including the Amur and Bengal, living in the wild. Scientists estimated that there were a total of 4,000 tigers in the wild. Of these, 450 were Amur tigers and x were Bengal tigers. Which inequality represents all possible numbers of tigers in the wild in 2008 that belonged to the Bengal tiger subspecies?
A. 5𝑥 ≥ 4000
B. $$\frac{x}{5}\geq 4000$$
C. 1 ≤ 𝑥 ≤ 3547
D. 3547 < 𝑥 ≤ 4000

Ans: C

To solve this problem, we need to find the range of possible values for the number of Bengal tigers (x) based on the given information.

• There were a total of 4,000 tigers in the wild.
• Out of these 4,000 tigers, 450 were Amur tigers.
• There were 5 known subspecies of tigers in the wild.

Since we know the total number of tigers and the number of Amur tigers, we can find the number of tigers belonging to the remaining 4 subspecies (including the Bengal tigers) by subtracting the number of Amur tigers from the total.

Number of tigers belonging to the remaining 4 subspecies = 4,000 – 450 = 3550

Since the Bengal tigers are one of the 4 remaining subspecies, the number of Bengal tigers (x) must be less than or equal to 3,550.

Also, the number of Bengal tigers cannot be negative or zero, as it is stated that they were one of the known subspecies living in the wild.

Therefore, the inequality that represents all possible numbers of tigers in the wild in 2008 that belonged to the Bengal tiger subspecies is:

$1 ≤ x ≤ 3550$

Among the given options, the correct answer is $C. 1 ≤ x ≤ 3547$.

[Calc]  Question  Medium

The function f is defined by f(x) =mx +b, where m and b are constants such that m ≥ 1 and -7 ≤b ≤7 Which of the following could be the graph of y =f(x) ?

Ans: C

[Calc]  Question   Medium

Teachers and students will participate in a tutoring seminar, and each teacher will lead a group of no more than 8 students. The room for the seminar can hold a maximum of 58 teachers and students. If t represents the number of teachers ands represents the number of students, which system of inequalities describes the possible numbers of teachers and students who can participate in the seminar ?
A) t + s ≥ 58
t≤ 8s

B) t + s ≤ 58
t ≥ 8s

C) t+ s ≥ 58

$$t ≥ \frac{s}{8}$$

D) t+s ≤ 58

$$t ≥ \frac{s}{8}$$

D) t+s ≤ 58

$$t ≥ \frac{s}{8}$$

Inequality 1: The total number of teachers and students cannot exceed 58. $t + s ≤ 58$

Inequality 2: The number of students for each teacher cannot exceed 8. $s ≤ 8t$

Combining the two inequalities, we get the system of inequalities: $t + s ≤ 58 s ≤ 8t$

Now, let’s check each option:

Option A: $t + s ≥ 58, t ≤ 8s$ This option is incorrect because the first inequality contradicts the given condition that the maximum number of teachers and students is 58.

Option B: $t + s ≤ 58, t ≥ 8s$ This option is incorrect because the second inequality implies that each teacher must have at least 8 students, which contradicts the given condition that each teacher will lead a group of no more than 8 students.

Option C: $t + s ≥ 58, t ≥ s/8$ This option is incorrect because the first inequality contradicts the given condition that the maximum number of teachers and students is 58.

Option D: $t + s ≤ 58, t ≥ s/8$ This option is correct because it represents the system of inequalities that satisfies both conditions:

1. The total number of teachers and students cannot exceed $58 (t + s ≤ 58)$.
2. The number of students for each teacher cannot exceed 8 $(t ≥ s/8,$ which can be rearranged as $s ≤ 8t)$.

Therefore, the correct answer is D)

[Calc]  Question  Medium

The shaded region shown represents the solutions to which inequality?

A. y≥-5x+3

B. y2-3x+5

C. y≤3x+5

D. y≤5x+3

Ans: D

From the graph value of y intercept is 3 so Option A and D are possible chances.

We can observe that the shaded region covers the area below the line passing through the points (0, 3) and (1, 8). The slope of this line is (8 – 3) / (1 – 0) = 5, which represents the coefficient of x in the inequality.

The y-intercept of the line is 3, which corresponds to the constant term in the inequality.

Therefore, the inequality y ≤ 5x + 3 accurately describes the shaded region, where all the points (x, y) below or on the line satisfy the inequality.

[Calc]  Question  medium

$$y<\frac{1}{2}x+4$$

$$y>-2x+4$$

Which ordered pair (x, y) is a solution to the given system of inequalities in the xy-plane?

A) (0,2)

B) (1,0)

C) (1,5)

D) (2,4)

D)  (2,4)

Given the system of inequalities:
\begin{align*} & y < \frac{1}{2}x + 4 \\ & y > -2x + 4 \end{align*}

We need to find which ordered pair $$(x, y)$$ is a solution to this system.

Let’s analyze each inequality:
$$y < \frac{1}{2}x + 4$$ represents a region below the line $$y = \frac{1}{2}x + 4$$.
$$y > -2x + 4$$ represents a region above the line $$y = -2x + 4$$.

We need to find the intersection of these two regions.

test each ordered pair to see which one satisfies both inequalities:

• A) $$(0,2)$$:
$$2 < \frac{1}{2}(0) + 4$$ is true (below the line $$y = \frac{1}{2}x + 4$$.
$$2 > -2(0) + 4$$ is true (above the line $$y = -2x + 4$$).
Both inequalities are satisfied, so option A is a solution.
• B) $$(1,0)$$:
$$0 < \frac{1}{2}(1) + 4$$ is false (above the line $$y = \frac{1}{2}x + 4$$.
$$0 > -2(1) + 4$$ is true (above the line $$y = -2x + 4$$).
Not a solution.
•  C) $$(1,5)$$:
$$5 < \frac{1}{2}(1) + 4$$ is false (above the line $$y = \frac{1}{2}x + 4$$.
$$5 > -2(1) + 4$$ is false (below the line $$y = -2x + 4$$).
Not a solution.
• D) $$(2,4)$$:
$$4 < \frac{1}{2}(2) + 4$$ is true (below the line $$y = \frac{1}{2}x + 4$$.
$$4 > -2(2) + 4$$ is true (above the line $$y = -2x + 4$$.
Both inequalities are satisfied, so option D is a solution.

Therefore, the solution is $$(2,4)$$.

[Calc]  Question   medium

In a forest, white pine trees between 15 and 45 years old grew 36 to 48 inches in height each year. A 15- year-old white pine tree growing in the forest was 240 inches tall. Which of the following inequalities gives all possible values for the tree’s height h, in inches, at the end of its 45th year?

A)h ≤ 540

B)h ≤ 2,160

C)240 ≤ h ≤ 1,080

D)1,320 ≤ h ≤ 1,680

D)1,320 ≤ h ≤ 1,680

To determine the possible values for the height $$h$$ of the white pine tree at the end of its 45th year, we need to consider its growth rate and current height.

Given:
The tree is currently 15 years old and 240 inches tall.
The growth rate is between 36 and 48 inches per year.
We want to find the height of the tree at the end of its 45th year (i.e., after 30 more years of growth).

Calculation:
1. Minimum Growth:

Growth per year: 36 inches
Number of years: 30
Total growth over 30 years: $$36 \, \text{inches/year} \times 30 \, \text{years} = 1080 \, \text{inches}$$

2. Maximum Growth:

Growth per year: 48 inches
Number of years: 30
Total growth over 30 years: $$48 \, \text{inches/year} \times 30 \, \text{years} = 1440 \, \text{inches}$$

3. Total Height Calculation:

Initial height at 15 years: 240 inches
Minimum possible height at 45 years: $$240 \, \text{inches} + 1080 \, \text{inches} = 1320 \, \text{inches}$$
Maximum possible height at 45 years: $$240 \, \text{inches} + 1440 \, \text{inches} = 1680 \, \text{inches}$$

The possible values for the tree’s height $$h$$ at the end of its 45th year lie between the calculated minimum and maximum heights.

Thus, the correct inequality is:D) $$1320 \leq h \leq 1680$$

[Calc]  Question  medium

The table gives the average speed s, in miles per hour (mph), of each lap around the track for one racing team. For how many laps was the average speed greater than or equal to 150 mph?

144

To find out how many laps had an average speed greater than or equal to $$150 \, \text{mph}$$, we need to sum up the number of laps for the speed intervals from $$150 \, \text{mph}$$ to $$165 \, \text{mph}$$.

From the table, we see that for the speed intervals $$150 \leq s < 155$$ , $$155 \leq s < 160$$ and $$160 \leq s < 165$$there are 57 and 52 laps  and 35 laps respectively.

$57 + 52+35 = 144$

So, for $$150 \, \text{mph} \leq s < 165 \, \text{mph}$$, there were $$144$$ laps with an average speed greater than or equal to $$150 \, \text{mph}$$.

[No- Calc]  Question   Medium

$y<x-4$

Which of the following ordered pairs $$(x, y)$$ satisfies the inequality above?
A) $$(0,3)$$
B) $$(3,0)$$
C) $$(0,6)$$
D) $$(6,0)$$

Ans:D

To determine which ordered pair $$(x, y)$$ satisfies the inequality $$y < x – 4$$, we can simply substitute the $$x$$ and $$y$$ values from each option into the inequality and see which one holds true.

A) $$(0, 3)$$
$y = 3$
$x – 4 = 0 – 4 = -4$
$3 < -4$
This is false.

B) $$(3, 0)$$
$y = 0$
$x – 4 = 3 – 4 = -1$
$0 < -1$
This is false.

C) $$(0, 6)$$
$y = 6$
$x – 4 = 0 – 4 = -4$
$6 < -4$
This is false.

D) $$(6, 0)$$
$y = 0$
$x – 4 = 6 – 4 = 2$
$0 < 2$
This is true.

So, the ordered pair that satisfies the inequality is:

D) $$(6, 0)$$

[Calc]  Question   Medium

\begin{aligned} & y<-3 x+1 \\ & y<-\frac{1}{2} x+1 \end{aligned}

Which ordered pair $(x, y)$ is a solution to the given system of inequalities in the $x y$-plane?
A) $(-2,3)$
B) $(1,2)$
C) $(0,2)$
D) $(-1,1)$

D

[Calc]  Question Medium

The coordinates of points $A, B$, and $C$ are shown in the $x y$-plane above. For which of the following inequalities will each of the points $A, B$, and $C$ be contained in the solution region?
A) $y>-x-2$
B) $y \geq-x$
C) $y<x+3$
D) $x<3$

A

[Calc]  Question  Medium

The function $f$ is defined for all real numbers, and the graph of $y=f(x)$ in the $x y$-plane is a line with a negative slope. Which of the following must be true?
I. If $a<b$, then $f(a)>f(b)$.
II. If $a<0$, then $f(a)>0$.
III. If $a>0$, then $f(a)<0$.
A) I only
B) II only
C) I and III only
D) II and III only

A

Question

A clothing store is having a sale on shirts and pants. During the sale, the cost of each shirt is $\$ 15$and the cost of each pair of pants is$\$25$. Geoff can spend at most $\$ 120$at the store. If Geoff buys$s$shirts and$p$pairs of pants, which of the following must be true? A.$15 s+25 p \leq 120$B.$15 s+25 p \geq 120$C.$25 s+15 p \leq 120$D.$25 s+15 p \geq 120$▶️Answer/Explanation Ans:A Question A certain elephant weighs 200 pounds at birth and gains more than 2 but less than 3 pounds per day during its first year. Which of the following inequalities represents all possible weights$w$, in pounds, for the elephant 365 days after its birth? A.$400<w<600$B.$565<w<930$C.$730<w<1,095$D.$930<w<1,295$▶️Answer/Explanation Ans: D Questions During an ice age, the average annual global temperature was at least 4 degrees Celsius lower than the modern average. If the average annual temperature of an ice age is$y$degrees Celsius and the modern average annual temperature is$x$degrees Celsius, which of the following must be true? A.$y=x-4$B.$y \leq x+4$C.$y \geq x-4$D.$y \leq x-4$▶️Answer/Explanation Ans: D Question Emma mows grass at a constant rate of 1.5 acres per hour. She mowed 2 acres before lunch and plans to spend$t$hours mowing after lunch. If Emma wants to mow at least 8 acres of grass today, which of the following inequalities best represents this situation? A.$1.5 t \geq 8$B.$1.5 t-2 \geq 8$C.$1.5 t+2 \geq 8$D.$2 t+1.5 t \geq 8$▶️Answer/Explanation Ans: C Questions A bag containing 10,000 beads of assorted colors is purchased from a craft store. To estimate the percent of red beads in the bag, a sample of beads is selected at random. The percent of red beads in the bag was estimated to be$15 \%$, with an associated margin of error of$2 \%$. If$r$is the actual number of red beads in the bag, which of the following is most plausible? A.$r>1,700$B.$1,300<r<1,700$C.$200<r<1,500$D.$r<1,300$▶️Answer/Explanation Ans: B Questions Terrence’s car contains 8 gallons of fuel. He plans to drive the car$m$miles using the fuel currently in the car. If the car can drive 20 miles per gallon of fuel, which inequality gives the possible values of$m$? A.$m \leq(8)(20)$B.$m \geq(8)(20)$C.$8 \leq 20 m$D.$8 \geq 20 m\$

Ans: A

Questions

During mineral formation, the same chemical compound can become different minerals depending on the temperature and pressure at the time of formation. A phase diagram is a graph that shows the conditions that are needed to form each mineral. The graph above is
a portion of the phase diagram for aluminosilicates, with the temperature $$T$$, in degrees Celsius (°C), on the horizontal axis, and the pressure $$P$$, in gigapascals (GPa), on the vertical axis.

Which of the following systems of inequalities best describes the region where sillimanite can form?

1. $$P \geq 0.0021T — 0.67$$
$$P \geq 0.0013T — 0.25$$
2. $$P \leq 0.0021T — 0.67$$
$$P \geq —0.0015T+ 1.13$$
3. $$P \leq 0.0013T— 0.25$$
$$P \geq —0.0015T+ 1.13$$
4. $$P \leq 0.0013T— 0.25$$
$$P \leq —0.0015T+ 1.13$$

Ans: B

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