Choose the missing step in the given solution to the inequality −x − 10 > 14 + 2x.
−x − 10 > 14 + 2x
−3x − 10 > 14
_____________
x < −8
−3x > 24
−3x > 4
−3x > −4
−3x < 24
Add 2x to both sides:
-x - 10 + 2x > 14
Simplify:
x - 10 > 14
Add 10 to both sides:
x > 24
The missing step is:
Divide both sides by -1 and flip the inequality symbol:
x < -24
Two boats are traveling in a canal in opposite directions, where x represents the distance, in feet, between the boats. The equation 4(x + 1) = 4x + 4 models the paths of the boats. Solve for x.
No solution
0
All real numbers
1
We can simplify the equation:
4(x + 1) = 4x + 4
Distribute the 4:
4x + 4 = 4x + 4
Simplify:
4 = 4
This equation is true for all values of x. Therefore, the solution is:
All real numbers.
The perimeter of a rectangle can be found using the equation P = 2L + 2W, where P is the perimeter, L is the length, and W is the width of the rectangle. Can the perimeter of the rectangle be 60 units when its width is 11 units and its length is 24 units?
No. If the length is 24 units and the width is 11 units, the perimeter would be P = 48 + 22 = 70 units, not 60.
No. If the length is 24 units and the width is 11 units, the perimeter would be P = 24 + 22 = 66 units, not 60.
Yes. If the perimeter is 60 units and the width is 11 units, then P + W is greater than 48.
Yes. If the length is 24 units and the width is 11 units, then P = 2L + 2W = 60
No. If the length is 24 units and the width is 11 units, the perimeter would be P = 2(24) + 2(11) = 48 + 22 = 70 units, not 60.
Zoya has to earn at least $300 to meet her fundraising goal. She has only 100 bracelets that she plans to sell at $5 each. Which inequality shows the number of bracelets, x, Zoya can sell to meet her goal?
60 ≤ x ≤ 100
60 ≤ x ≤ 300
100 ≤ x ≤ 300
20 ≤ x ≤ 100
Zoya plans to sell each bracelet at $5, so the total amount she would earn by selling x bracelets would be 5x.
To meet her goal, she has to earn at least $300:
5x ≥ 300
Divide both sides by 5:
x ≥ 60
Therefore, Zoya has to sell at least 60 bracelets to meet her goal.
The correct answer is:
60 ≤ x ≤ 100 (since she only has 100 bracelets to sell).
Solve for x: −6 < x − 1 < 9
5 < x < 10
−5 < x < 10
−5 > x > 10
5 > x > −10
Add 1 to all parts of the inequality:
-6 + 1 < x - 1 + 1 < 9 + 1
Simplify:
-5 < x < 10
The correct answer is:
-5 < x < 10.
To find the missing step in the given solution to the inequality −x − 10 > 14 + 2x, let's go through the steps and identify where the missing step should be inserted:
−x − 10 > 14 + 2x
−3x − 10 > 14 (This step is correct and shows the distribution of -3 on both terms)
−3x > 24 (To isolate the variable, we need to add 10 to both sides)
_________
x < −8 (This step is correct and shows the final solution)
The missing step should be inserted after the step "-3x > 24" to further simplify the inequality.
To find the missing step, consider that we have a negative coefficient (-3x) on the left side of the inequality. If we divide both sides by a negative number (-3), the inequality sign will flip. So, dividing by -3, we get:
(-3x) / (-3) < 24 / (-3)
This simplifies to:
x > -8
So, the missing step is:
-3x > 24
-3x / (-3) < 24 / (-3)
x > -8
Therefore, the solution to the inequality is x > -8.