Consider the following expressions: 6−14⋅28−214 _______ 34+4⋅2−11 . Which operator can be inserted to make the statement true? (1 point) Responses = equals > greater than < less than ≥ greater-than-or-equal-to
To make the statement true, we need to determine the correct operator to insert. Comparing the two expressions, we see that 6 - 14 ⋅ 28 - 214 is being compared to 34 + 4 ⋅ 2 - 11.
The value of 6 - 14 ⋅ 28 - 214 is -392, and the value of 34 + 4 ⋅ 2 - 11 is 37.
Since -392 is less than 37, the correct operator to insert is < (less than).
Which of the following values for b makes the inequality −32.5−6b≥48.5 true?(1 point) Responses −13 negative 13 −12 negative 12 −13.5 negative 13.5 −12.5 negative 12.5
To solve the inequality -32.5 - 6b ≥ 48.5, we need to isolate the variable b.
First, let's add 32.5 to both sides of the inequality:
-32.5 + 32.5 - 6b ≥ 48.5 + 32.5
Simplifying:
-6b ≥ 81
Next, we divide both sides of the inequality by -6. It's important to remember that when we divide or multiply both sides of the inequality by a negative number, we must reverse the inequality sign.
-6b/(-6) ≤ 81/(-6)
Simplifying:
b ≤ -13.5
The inequality is true for any value of b that is less than or equal to -13.5.
Therefore, the correct response is: -13.5 negative 13.5
Which of the following equations has the solution set {−3} ?(1 point) Responses 4x+8=−4 4 x plus 8 equals negative 4 4x−8=4 4 x minus 8 equals 4 4x−8=−4 4 x minus 8 equals negative 4 4x+8=4 4 x plus 8 equals 4
To find which equation has the solution set {-3}, we can substitute -3 for the variable x in each equation and see which one satisfies the equation.
1) 4x+8=-4
Substituting -3 for x:
4(-3) + 8 = -4
-12 + 8 = -4
-4 = -4
This equation is not satisfied by the solution -3, so it is not the correct equation.
2) 4x-8=4
Substituting -3 for x:
4(-3) - 8 = 4
-12 - 8 = 4
-20 = 4
This equation is also not satisfied by the solution -3, so it is not the correct equation.
3) 4x-8=-4
Substituting -3 for x:
4(-3) - 8 = -4
-12 - 8 = -4
-20 = -4
This equation is not satisfied by the solution -3, so it is not the correct equation.
4) 4x+8=4
Substituting -3 for x:
4(-3) + 8 = 4
-12 + 8 = 4
-4 = 4
This equation is not satisfied by the solution -3, so it is not the correct equation.
Based on the above analysis, none of the provided equations has the solution set {-3}.
Which set expresses the solution to the compound inequality −2≤2x+13−4<1 ?(1 point) Responses {x:−32≤x<1} left brace x colon Start Fraction negative 3 over 2 End Fraction less-than-or-equal-to x less than 1 right brace {x:52≤x<1} left brace x colon Start Fraction 5 over 2 End Fraction less-than-or-equal-to x less than 1 right brace {x:52≤x<7} left brace x colon Start Fraction 5 over 2 End Fraction less-than-or-equal-to x less than 7 right brace {x:3≤x<7} left brace x colon 3 less-than-or-equal-to x less than 7 right brace
To solve the compound inequality -2 ≤ 2x + 13 - 4 < 1, we need to isolate the variable x.
First, let's simplify the expression:
-2 ≤ 2x + 13 - 4 < 1
Combine like terms:
-2 ≤ 2x + 9 < 1
Next, let's isolate 2x:
-2 - 9 ≤ 2x < 1 - 9
Simplifying:
-11 ≤ 2x < -8
Now, we can divide all three parts of the inequality by 2 (since we want to isolate x):
-11/2 ≤ 2x/2 < -8/2
Simplifying:
-11/2 ≤ x < -4
Therefore, the correct response is: {x: -11/2 ≤ x < -4} or in decimal form, {x: -5.5 ≤ x < -4}.
What property of equality is displayed in step 3 in the solving of 9m+4−7m=5m+73 ? Step 1: 9m+4−7m=5m+73 9m−7m+4=5m+73 Step 2: 2m+4=5m+73 3(2m+4)=(5m+73)3 Step 3: 6m+12=5m+7 6m+12−12=5m+7−12 Step 4: 6m=5m−5 6m−5m=5m−5m−5 m=−5 (1 point) Responses Addition Property of Equality Addition Property of Equality Multiplication Property of Equality Multiplication Property of Equality Division Property of Equality Division Property of Equality Subtraction Property of Equality Subtraction Property of Equality
The property of equality displayed in Step 3 of the solving process is the Subtraction Property of Equality.
In Step 2, we have 6m + 12 = 5m + 7. To isolate the variable m, we subtract 12 from both sides of the equation in Step 3.
6m + 12 - 12 = 5m + 7 - 12
Simplifying:
6m = 5m - 5
So, the Subtraction Property of Equality is used in Step 3.
Which linear equation has the same solution set as 9d4+3−2d=−2(3+d) ?(1 point) Responses 35−15b=95 35 minus 15 b equals 95 14=74x−194+2x 14 equals Start Fraction 7 over 4 End Fraction x minus Start Fraction 19 over 4 End Fraction plus 2 x 4w−2(7−3w)=w3−6 4 w minus 2 left parenthesis 7 minus 3 w right parenthesis equals Start Fraction w over 3 End Fraction minus 6 3x5−10=−7
To find the linear equation that has the same solution set as 9d4+3−2d=−2(3+d), we need to simplify the given equation and determine the equivalent form.
Starting with the given equation:
9d4 + 3 - 2d = -2(3 + d)
Start by distributing the -2 on the right side:
9d4 + 3 - 2d = -6 - 2d
Combine like terms:
9d - 2d + 3 = -6 - 2d
Simplify further:
7d + 3 = -6 - 2d
To find the equivalent equation, we need to manipulate the equation to isolate the variable d on one side.
Add 2d on both sides:
7d + 2d + 3 = -6 - 2d + 2d
Simplifying:
9d + 3 = -6
Subtract 3 on both sides:
9d + 3 - 3 = -6 - 3
Simplifying:
9d = -9
Divide both sides by 9:
Start Fraction 9 d Over 9 End Fraction = Start Fraction negative 9 Over 9 End Fraction
Simplifying:
d = -1
So, the linear equation that has the same solution set as 9d4+3−2d=−2(3+d) is d = -1.
Which option is the solution to the inequality −12g+18>−5g+53 ?(1 point) Responses g<71−17 g less than Start Fraction 71 over negative 17 End Fraction g<−5 g less than negative 5 g>71−17 g greater than Start Fraction 71 over negative 17 End Fraction g>−5
To find the solution to the inequality -12g + 18 > -5g + 53, we need to isolate the variable g.
Start by subtracting -18 from both sides of the inequality:
-12g + 18 - 18 > -5g + 53 - 18
-12g > -5g + 35
Next, subtract -5g from both sides of the inequality:
-12g - (-5g) > -5g - (-5g) + 35
-12g + 5g > 0 + 35
Simplifying:
-7g > 35
To isolate g, divide both sides of the inequality by -7. Remember that dividing (or multiplying) by a negative number when solving an inequality will reverse the inequality sign.
-7g/(-7) < 35/(-7)
Simplifying:
g < -5
Therefore, the solution to the inequality -12g + 18 > -5g + 53 is g < -5.
The correct response is: g < -5.
Find the solution set of 4x−3>9 and x+6>−2 .(1 point) Responses {x|x>3} left brace x vertical bar x greater than 3 right brace {x|x>−8} left brace x vertical bar x greater than negative 8 right brace {3,−8} left brace 3 comma negative 8 right brace The solution set is null.
To find the solution set of the inequalities 4x - 3 > 9 and x + 6 > -2, we solve each inequality separately.
For 4x - 3 > 9:
Add 3 to both sides:
4x - 3 + 3 > 9 + 3
4x > 12
Divide both sides by 4 (since the coefficient of x is 4 and we want to isolate x):
x > 3
For x + 6 > -2:
Subtract 6 from both sides:
x + 6 - 6 > -2 - 6
x > -8
The solution set for the given system of inequalities is {x | x > 3, x > -8}. However, since both inequalities require x to be greater than 3, the solution set can be simplified to just {x | x > 3}.
Therefore, the correct response is: {x | x > 3}.
Use the image to answer the question. Select the solution set represented on the graph. (1 point) Responses x≤−3 or x>1 x less-than-or-equal-to negative 3 or x greater than 1 x<−3 or x>1 x less than negative 3 or x greater than 1 x≤−3 or x≥1 x less-than-or-equal-to negative 3 or x greater-than-or-equal-to 1 x<−3 and x>1
Since the image representing the solution set is not provided, I am unable to determine which option correctly represents the solution set on the graph.
What is the undefined value for the equation 14=7x9+4x ?(1 point) Responses x≠−94 x does not equal Start Fraction negative 9 over 4 End Fraction x≠94 x does not equal Start Fraction 9 over 4 End Fraction x≠−49 x does not equal Start Fraction negative 4 over 9 End Fraction x≠0
To find the undefined value for the equation 14 = 7x9 + 4x, we need to determine the value of x that would make the equation undefined.
Starting with the equation 14 = 7x9 + 4x, we can simplify it:
14 = 63x + 4x
Combining like terms:
14 = 67x
To isolate x, we need to divide both sides of the equation by 67:
14/67 = x
This simplifies to:
x ≈ 0.208
Therefore, the undefined value for the equation 14 = 7x9 + 4x is x ≠ 0.
Which correctly shows the area formula for a trapezoid, A=12h(b1+b2) , rearranged for the quantity of interest h?(1 point) Responses h=2Ab1+b2 h equals Start Fraction 2 upper A over b subscript 1 baseline plus b subscript 2 baseline End Fraction h=b1+b22A h equals Start Fraction b subscript 1 baseline plus b subscript 2 baseline over 2 upper A End Fraction b1=2Ah−b2 b subscript 1 baseline dequals Start Fraction 2 upper A over h End Fraction minus b subscript 2 baseline h=12A(b1+b2)
To rearrange the area formula for a trapezoid, A = 1/2h(b1 + b2), for the quantity of interest h, we need to isolate h.
Start with the given formula:
A = 1/2h(b1 + b2)
Multiply both sides of the equation by 2 to remove the fraction:
2A = h(b1 + b2)
Now, divide both sides of the equation by (b1 + b2) to isolate h:
2A / (b1 + b2) = h
Therefore, the correct response is h = 2A / (b1 + b2).
Given the formula for time in terms of distance and velocity, t=dv , what operation would you use to isolate the distance?(1 point) Responses addition addition division division subtraction subtraction multiplication
To isolate the distance, we need to rearrange the formula t = dv.
To do this, we need to "undo" the operation of multiplication by velocity (v). Since multiplication is the inverse operation of division, we would use the operation of division to isolate the distance (d).
Therefore, the correct operation to use to isolate the distance is division.
Sebastian needed to buy new tires for his car. The tire shop was running a holiday deal that allowed customers to buy three tires and get the fourth tire for $5. If the total price for all his tires was $237.50, what was the cost of each of the first three tires?(1 point) Responses $59.38 $59.38 $79.17 $79.17 $77.50 $77.50 $80.83 $80.83
Let's assume the cost of each of the first three tires is x.
Since the deal allows customers to buy three tires and get the fourth for $5, the cost of the fourth tire is $5.
Sebastian bought three tires at the cost of x each, so the cost of those three tires is 3x.
So, the total price for all his tires, including the fourth one, is:
3x + $5.
We're given that the total price for all his tires is $237.50, so we can set up the equation:
3x + $5 = $237.50.
Subtracting $5 from both sides gives:
3x = $237.50 - $5,
3x = $232.50.
To find the cost of each of the first three tires (x), we need to divide both sides of the equation by 3:
x = $232.50 / 3,
x ≈ $77.50.
Therefore, the cost of each of the first three tires is approximately $77.50.
The correct response is: $77.50.
A 10-gallon water cooler in an office provides water for the whole department. Each hour, 30 ounces of water are removed from the cooler and drunk by office workers. Write an equation to show how long the water in the cooler will last. (10 gallons is 1,280 ounces.)(1 point) Responses 1,280−30h=0 1,280 minus 30 h equals 0 10−30h=0 10 minus 30 h equals 0 1,280+30h=0 1,280 plus 30 h equals 0 30h=10 30 h equals 10
To write an equation to show how long the water in the cooler will last, we need to consider the rate at which water is removed from the cooler.
Each hour, 30 ounces of water are removed.
Let's denote the time in hours as h.
The initial amount of water in the cooler is 1,280 ounces (10 gallons is 1,280 ounces).
Thus, the equation would be:
1,280 - 30h = 0
This equation represents the amount of water remaining in the cooler (1,280 ounces) minus the amount of water removed after h hours (30h) equals 0, indicating that there is no water left in the cooler.
Therefore, the correct equation is: 1,280 - 30h = 0.
Dezeree received a $100 gift card and needs new sweaters for school. The sweaters cost $27.50 each. Write an equation to determine how many sweaters she can buy and spend exactly $100, solve it, and determine if the value is a viable solution.(4 points)
To determine how many sweaters Dezeree can buy and spend exactly $100, we can write an equation.
Let's denote the number of sweaters she can buy as x.
The cost of each sweater is $27.50.
The equation to represent the total cost of the sweaters is:
27.50x = 100
To solve the equation, divide both sides by $27.50:
x =