UNIT: SOLVING EQUATIONS AND INEQUALITIES
The solution is n = -2 verified as a solution to the equation 1.4n + 2 = 2n + 3.2. What is the last line of the justification?
-0.8 = -0.8
What is a correct first step in solving the inequality -4(3 - 5x)≥ -6x + 9?
-12 + 20x ≥ -6x + 9
Jeremy rides a Ferris wheel. The graph shows h, Jeremy's height above the ground at any time during the ride. Which inequality represents the heights shown on the graph?
10 ≤ h ≤ 50
Last week, Lindsay earned $10 per hour plus a $60 bonus for good job performance. She spends 1/15 of her paycheck on dinner with friends. If she had not earned the bonus, the amount she spent on dinner would have been 1/10 of her paycheck. Which equation can be used to find h, the number of hours Lindsay worked last week?
1/15(10h + 60) = 1/10(10h)
Sara wants to find the input value that produces the same output for the functions represented by the tables. What is the input value that produces the same output value in both charts?
2
Consider the equation 5/3v + 4 + 1/3v = 8. What is the resulting equation after the first step in the solution?
2v + 4 = 8
Mrs. Ishimitsu is installing a rubber bumper around the edge of her coffee table. The dimensions of the rectangular table are (2x2 - 16) feet and (-x2 + 4x + 1) feet. Which expression represents the total perimeter of the table, and if x = 3, what is the length of the entire rubber bumper?
2x2 + 8x - 30; 12 feet
Jill simplified the expression shown. 4x - 2y - x + 5y - 8 What is the coefficient of the x-term of the fully simplified expression?
3
Which is a correct first step in solving 5 - 2x < 8x - 3?
5 < 10x - 3
Sianna bought 3 pens, 4 notebooks, and a calculator. The calculator cost $25. If x represents the cost of each pen and y represents the cost of each notebook, which expression represents the amount of change Sianna should receive from $50?
50 - (3x + 4y + 25)
For one photography session, Dexter earns no less than $50, but no more than $100. Which inequality can be used to represent his earnings, e?
50 ≤ e ≤ 100
A political polling agency predicts candidate A will win an election with 54% of the votes. Their poll has a margin of error of 4% both above and below the predicted percentage. Which inequality represents the predicted possible percent of votes, x, for candidate A?
50 ≤ x ≤ 58
What value of x is in the solution set of -2(3x + 2) > -8x + 6?
6
Which equation can be used to represent "six added to twice the sum of a number and four is equal to one-half of the difference of three and the number"?
6 + 2(x + 4) = 1/2(3 - x)
The sum of the interior angles, s, in an n-sided polygon can be determined using the formula s = 180(n - 2), where n is the number of sides. Benita solves this equation for n and writes the equivalent equation n = s/180 + 2. Using this formula, how many sides does a polygon have if the sum of the interior angles is 1,260°?
9 sides
Hiroshi spends 30 minutes on history homework, 60 minutes on English homework, and x minutes on math homework. One fourth of his total homework time is spent on math. Which equation can be used to find the amount of time Hiroshi spends on his math homework?
A
Step 1: Subtract 3 from both sides of the inequality. Step 2: ___________ Step 3: Divide both sides of the inequality by the coefficient of x. What is the missing step in solving the inequality 5 - 8x < 2x + 3?
Add 8x to both sides of the inequality.
The goals against average (A) for a professional hockey goalie is determined using the formula A = 60 . In the formula, g represents the number of goals scored against the goalie and t represents the time played, in minutes. Which is an equivalent equation solved for g?
At/60 = g
The goals against average (A) for a professional hockey goalie is determined using the formula A = 60g/t . In the formula, g represents the number of goals scored against the goalie and t represents the time played, in minutes. Which is an equivalent equation solved for g?
At/60 = g
The amount of potential energy, P, an object has is equal to the product of its mass, m, its height off the ground, h, and the gravitational constant, g. This can be modeled by the equation P = mgh. What is the equivalent equation solved for h?
B
The axis of symmetry for a quadratic equation can be found using the formula x = -b/2a , where a and b are coefficients in the quadratic equation and x represents the values along a vertical line on the coordinate plane. When the equation is solved for a, such that b is the numerator of the resulting fraction, what is the denominator of the fraction?
B
Which description of the graph of the linear equality y > 3x - 8 is correct?
D
Which linear inequality is represented by the graph? y > 2x + 3 y < 2x + 3 y > −2x + 3 y < −2x + 3
D
Which equation has only one solution? |x - 5| = -1 |-6 - 2x| = 8 |5x + 10| = 10 |-6x + 3| = 0
D. |-6x + 3| = 0
Bianca wrote the steps for her solution to the equation 2.3 + 8(1.3x - 4.75) = 629.9. She left out the description for the last step. What is the missing step of her solution?
Divide both sides of the equation by 10.4.
Marta is solving the equation S= 2πrh + 2πr2 for h. Which should be the result?
S/2πr -r = h
Tasha earns $400 per week plus a commission of 10% on her weekly sales. Each week she saves of her earnings. In the expression , what does the expression 400 + 0.1s represent in this situation?
Tasha's weekly earnings
If g(x) = 2[x] − 1, what is g(−2.3)? a. -7 b. -5 c. -4 d. -3
a. -7
Four milliliters of ink A is added to 10 milliliters of ink B, forming a mixture of inks. Ink A contains 10% blue pigment. Ink B contains an unknown percentage of blue pigment. The mixture of the two inks contains 35% blue pigment. What percentage of blue pigment is in ink B? a. 45% b. 49% c. 60% d. 75%
a. 45%
Tracie rides the bus home from school each day. The graph represents her distance from home relative to the number of minutes since the bus left the school. What does the slope of the graph mean? a. Tracie's bus travels towards her home at an average speed of 1/2 mile per minute. b. Tracie's bus travels towards her home at an average speed of 2 miles per minute. c. Tracie's bus travels away from her home at an average speed of 1/2 mile per minute. d. Tracie's bus travels away from her home at an average speed of 2 miles per minute.
a. Tracie's bus travels towards her home at an average speed of 1/2 mile per minute.
Johan found that the equation -2|8 - x| - 6 = -12 had two possible solutions: x = 5 and x = -11. Which explains whether his solutions are correct? He is correct because both solutions satisfy the equation. He is not correct because he made a sign error. He is not correct because there are no solutions. He is not correct because there is only one solution: x = 5.
b
The weight of a box of cereal can vary by 1/4 of an ounce and still be sold as a full box. Each box is supposed to contain 18 ounces of cereal. Which graph represents the possible weights of boxes that are overfilled or underfilled?
b
if f(x)=1/2 [x] what is f(7.6) a. 3 b. 3.5 c. 3.8 d. 4
b. 3.5
In order to avoid heavy traffic, Antonio can drive home after working less than 7 hours (h < 7) or after working more than 9 hours (h >9). How can these inequalities be written as one statement?
h < 7 or h > 9
Sharon's turtle escaped from her backyard sometime in the last few hours. According to her calculations, the farthest the turtle could have gone is 4 blocks down the road in either direction. If Sharon lives on the 112th block of town, which equation can be used to find the block numbers that represent the farthest distance that the turtle may be? |x - 112| = 0 |x - 112| = 4 |4 - 112| = x |112 - 4| = x
not c
In order to be invited to tutoring or extension activities, a student's performance must be at least 20 points higher or lower than an average score. If the average score is 105, which inequality represents p, the scores for students who are eligible for tutoring or extension activities?
p ≤ 85 or p ≥ 125
If a bus driver leaves her first stop by 7:00 a.m., her route will take less than 37 minutes. If she leaves after 7:00 a.m., she estimates that the same route will take no less than 42 minutes. Which inequality represents the time it takes to drive the route, r?
r < 37 or r ≥ 42
The formula for the lateral area of a right cone is LA = πrs, where r is the radius of the base and s is the slant height of the cone. Which are equivalent equations? Check all that apply.
s = LA / πr r = LA / πs
The equation f = v + at represents the final velocity of an object, f, with an initial velocity, v, and an acceleration rate, a, over time, t. Which is an equivalent equation solved for t?
t = f-v/a
Ahmed has five more CDs than one-half the number of CDs Julia has. In this situation, what does c+ 5 represent?
the number of CDs Ahmed has
The area, a, of an ellipse can be determined using the formula a = πxy, where x and y are half the lengths of the largest and smallest diameters of the ellipse. Which is an equivalent equation solved for y?
y = a / (πx)
Which set of numbers is included in the solution set of the compound inequality?
{-7, 5, 18, 24, 32}
Which set of numbers is included in the solution set of the compound inequality? {-7, 5, 18, 24, 32} {-9, 7, 15, 22, 26} {16, 17, 22, 23, 24} {18, 19, 20, 21, 22}
{-7, 5, 18, 24, 32}
In physics class, Carrie learns that a force, F, is equal to the mass of an object, m, times its acceleration, a. She writes the equation as F = ma, then realizes that each of her homework problems asks her to solve for acceleration. She rewrites the formula as f/m = a. Using this formula, what is the acceleration of an object with F = 7.92 newtons and m = 3.6 kilograms? Express your answer to the nearest tenth.
2.2 m/s2
To fit in an existing frame, the length, x, of a piece of glass must be longer than 12 cm but not longer than 12.2 cm. Which inequality can be used to represent the lengths of the glass that will fit in the frame?
12 < x ≤ 12.2
An inequality represents the ages, a, of players on a baseball team. Which inequality represents the situation?
12 ≤ a < 18
The inequality graphed below represents the ages, a, of players on a baseball team. Which inequality represents the same ages?
12 ≤ a < 18
y = 5x + 2 3x = -y + 10 What is the solution to the system of equations?
D
Which statement is true? The equation -3|2x + 1.2| = -1 has no solution. The equation 3.5|6x - 2| = 3.5 has one solution. The equation 5|-3.1x + 6.9| = -3.5 has two solutions. The equation -0.3|3 + 8x| = 0.9 has no solution.
D. The equation -0.3|3 + 8x| = 0.9 has no solution.
Which correct first step in solving the inequality -4(2x - 1) > 5 - 3x?
Distribute -4 to get -8x + 4 > 5 - 3x.
Which is a correct first step in solving the inequality -4(2x - 1) > 5 - 3x?
Distribute -4 to get -8x + 4 > 5 - 3x.
The formula for the circumference of a circle is C = 2πr, where r is the radius and C is the circumference. The equation solved for r is r =c/π2 . Find the radius of a circle that has a circumference of 16.
r = 8
The circumference of a circle can be found using the formula C = 2πr. Which is an equivalent equation solved for r?
r = C/2π
Which compound inequality is represented by the graph? -3 ≤ x < 1 -3 < x ≤ 1 x ≤ -3 or x > 1 x < -3 or x ≥ 1
-3 ≤ x < 1
Which are correct representations of the inequality -3(2x - 5) < 5(2 - x)? Select two options.
-6x + 15 < 10 - 5x and number line 1 (or, answers C and D)
Use properties to rewrite the given equation. Which equations have the same solution as 2.3p - 10.1 = 6.5p - 4 - 0.01p? Select two options.
2.3p - 10.1 = 6.49p - 4 230p - 1010 = 650p - 400 - p
What value of x is in the solution set of 2x - 3 > 11 - 5x?
4
The volume, V, of a rectangular prism is determined using the formula , where l is the length, w is the width, and h is the height of the prism. Carltren solves for w and writes the equivalent equation w = V/lh. Using this formula, what is the width of a rectangular prism that has a volume of 138.24 cubic inches, a height of 9.6 inches, and a length of 3.2 inches?
4.5 inches
Mr. Valenzuela's new bottle of shampoo contains 60 ounces of shampoo. He uses ounce of shampoo each time he washes his hair. Which equation can be used to find x, the number of washes after which 24 ounces will remain in the bottle?
60 - 1/4X =24
Lukas graphed the system of equations shown. 2x + 3y = 2 y = x + 3 What is the solution to the system of equations?
A
The formula for the area of a triangle is A =1/2 bh, where b is the length of the base and h is the height. The equation solved for h is h = 2a/b . Find the height of a triangle that has an area of 30 square units and a base measuring 12 units.
B
The table can be used to determine the solution of equations, 2x − 2y = 6 and 4x + 4y = 28. Which solution can be used to fill in both blanks in the table?
B
Sharon's turtle escaped from her backyard sometime in the last few hours. According to her calculations, the farthest the turtle could have gone is 4 blocks down the road in either direction. If Sharon lives on the 112th block of town, which equation can be used to find the block numbers that represent the farthest distance that the turtle may be?
B. |x - 112| = 4
A system of equations has infinitely many solutions. If 2y - 4x = 6 is one of the equations, which could be the other equation?
C
The solution to the system of equations shown is (2, 0). 3x − 2y = 6 x + 4y = 2 When the first equation is multiplied by 2, the sum of the two equations is equivalent to 7x = 14 . Which system of equations will also have a solution of (2, 0)?
C
The temperature measured in Kelvin (K) is the temperature measured in Celsius (C) increased by 273.15. This can be modeled by the equation K = C + 273.15. When solved for C, the equation is
C = K - 273.15
Morgan is walking her dog on an 8-meter-long leash. She is currently 500 meters from her house, so the maximum and minimum distances that the dog may be from the house can be found using the equation |x - 500| = 8. What are the minimum and maximum distances that Morgan's dog may be from the house?
C. 492 meters and 508 meters
For water to be a liquid, its temperature must be within 50 Kelvin of 323 Kelvin. Which equation can be used to determine the minimum and maximum temperatures between which water is a liquid?
C. |x - 323| = 50
Timmy writes the equation f(x)= 1/4x-1. he then doubles both of the terms on the right side to create the equation g(x)= 1/2x-2. how does the graph of g(x) compare to the graph of f(x)?
The line of g(x) is steeper and has a lower y-intercept
Which statements are true about the graph of the system of linear inequalities? Check all that apply. y > 3x - 4 y<=1/2(x+1) a. the graph of y > 3x − 4 has shading above a dashed line. b. the graph of y<=1/2 (x + 1) has shading below a dashed line. c. the graphs of the inequalities will intersect. d. there are no solutions to the system. e. the graphs of the two inequalities intersect the y-axis at (0, 1) and (0, 4).
a and c
The inequality x < 9 or x ≥ 14 can be used to represent the hourly wage, x, of each employee at a store. Which are possible values for x? Check all that apply. a. $8 b. $9 c. $11 d. $13 e. $14 f. $15
a, e, and f
To win a contest, the number of beans in a jar has to be guessed within 20 of the actual number. If the number of beans in the jar is 645, which equation can be used to find the minimum and maximum number of beans that will win the contest, and what is the maximum guess that could win? |x - 645| = 20; maximum guess: 655 beans |x - 645| = 20; maximum guess: 665 beans |x - 20| = 645; maximum guess: 655 beans |x - 20| = 645; maximum guess: 665 beans
b
For water to be a liquid, its temperature must be within 50 Kelvin of 323 Kelvin. Which equation can be used to determine the minimum and maximum temperatures between which water is a liquid? |323 - 50| = x |323 + 50| = x |x - 323| = 50 |x + 323| = 50
c
To make a small vase, Elisa uses no more than 4.5 ounces of clay. To make a large vase, she uses at least 12 ounces of clay. Which compound inequality represents the number of ounces of clay, c, that Elisa uses to make one vase of either size?
c ≤ 4.5 or c ≥ 12
Tickets to a school production cost $5 for a student ticket and $10 for an adult ticket. A total of 67 tickets were purchased at a cost of $440. Which value could replace c in the table? a. 67 b. 440 c. 67 - a d. 440 - a
c. 67 - a
Which values are solutions to (k-3)/(4)>-2 Check all that apply. a. k = -10 b. k = -7 c. k = -5 d. k = -1 e. k = 0 f. k = 5 g. k = 8
d, e, f, and g
which compound inequality could be represented by the graph? <--|---|---|---|---|---|---|---|---|---|---|--> -5 -4 -3 -2 -1 0 1 2 3 4 5 a. -4 ≤ x ≤ 4 b. -2 ≤ x ≤ -1 c. x ≤ -1 or x ≥ 0 d. x ≤ 3 or x ≥ -1
d. x ≤ 3 or x ≥ -1
The equation U=mgh represents the Gravitational energy of an object. U is the gravitational energy of an object, mi is its mass, g is the acceleration due to gravity, and h is the objects height. Which is an equivalent equation solved for h?
h=U/mg
A chef knows that ground beef used for hamburger is safe to eat once it reaches 160 degrees Fahrenheit. For optimized taste and safety, he will not allow a hamburger cooked to a temperature under 160 degrees or over 170 degrees to be served. Which graph represents the conditions under which the chef will not allow the hamburger to be served?
line 1
To support a shelf, the diameter of a bolt can measure anywhere from 0.2 cm less than 1.5 cm to 0.2 cm greater than 1.5 cm. Which graph represents the possible measures of the diameter of the bolt?
line 1
The weight of a box of cereal can vary by 1/4 of an ounce and still be sold as a full box. Each box is supposed to contain 18 ounces of cereal. Which graph represents the possible weights of boxes that are overfilled or underfilled and cannot be sold as full boxes?
line 2
The slope-intercept form of a linear equation is y = mx + b, where x and y are coordinates of an ordered pair, m is the slope of the line, and b is where the line crosses the y-axis. Which is an equivalent equation solved for the slope, m?
m = y-b/x
The equation d = m/v can be used to caculate the density, d, of an object with mass, m, and volume, V. Which us an equivalent equation solved for V?
m/d = V
The equation d = m/v can be used to calculate the density, d, of an object with mass, m, and volume, V. Which is an equivalent equation solved for V?
m/d = V
Laurie is trying to stay within 10 feet of her current diving depth of -30 feet (with regard to sea level) so that the light is still good but she can be close to the sea life during her scuba dive. Which two equations can be used to find the minimum and maximum depths Laurie wants to stay between?
x + 30 = 10 and x + 30 = -10
On the freeway, it is unsafe to drive too slowly or too quickly. The safest driving speeds on the freeway near Allan's town were scientifically determined to be between the minimum and maximum speeds (in miles per hour) according to the equation |x - 55| = 4.2. Which equation can be used to determine the maximum safe driving speed?
x - 55 = 4.2
Which compound inequality is represented by the graph? -2 < x ≤ 4 -2 ≤ x < 4 x < -2 or x ≥ 4 x ≤ -2 or x > 4
x < -2 or x ≥ 4
Step 1: -10 + 8x < 6x - 4 Step 2: -10 < -2x - 4 Step 3: -6 < -2x Step 4: ____________ What is the final step in solving the inequality -2(5 - 4) < 6x - 4?
x < 3
In order to qualify for a role in a play, an actor must be taller than 64 inches but shorter than 68 inches. The inequality 64 < x < 68, where x represents height, can be used to represent the height range. Which is another way of writing the inequality?
x > 64 and x < 68
Manuela solved the equation 3−2|0.5x+1.5|=2 for one solution. Her work is shown below. What is the other solution to the equation?
x=−4
The slope, m, of a linear equation can be found using the formula m= y2 - y1/ x2 - x1, where the x- and y-values come from two ordered pairs, and (x1, y1) and (x2, y2). What is an equivalent equation solved for y2?
y2 = m(x2 - x1) + y1