Choose the correct solution and graph for the inequality. z - 6 < - 14
The correct solution for the inequality is z < -8, and the graph of the solution is an open circle at -8 on the number line, shaded to the left.
To solve the inequality z - 6 < -14, you can follow these steps:
Step 1: Add 6 to both sides of the inequality to isolate the variable z:
z - 6 + 6 < -14 + 6
z < -8
Step 2: Graph the solution on a number line. Draw an open circle on -8 and shade the line to the left, indicating that z is less than -8.
So, the correct solution is z < -8, and the graph would be represented as:
----<=======o
Choose the correct solution and graph for the inequality.
y - 1/3 > 1/8
(1 point)
To solve the inequality y - 1/3 > 1/8, you can follow these steps:
Step 1: Add 1/3 to both sides of the inequality to isolate the variable y:
y - 1/3 + 1/3 > 1/8 + 1/3
y > 1/8 + 1/3
Step 2: Add the fractions 1/8 and 1/3:
1/8 + 1/3 = 3/24 + 8/24 = 11/24
So the inequality becomes:
y > 11/24
Step 3: Graph the solution on a number line. Draw an open circle on 11/24 and shade the line to the right, indicating that y is greater than 11/24.
The correct solution is y > 11/24, and the graph would be represented as:
-------o========>
To solve the inequality z - 6 < -14, we need to isolate the variable z.
Here are the steps to solve the inequality:
1. Add 6 to both sides of the inequality:
z - 6 + 6 < -14 + 6
Simplifying:
z < -8
Now, let's graph the solution on the number line:
1. Draw a number line with a point at -8 (since z < -8).
2. Since the inequality is not inclusive, we will leave an open circle at -8.
3. Shade the portion of the number line to the left of -8, indicating that z can be any value less than -8.
The graph should represent all values of z that are less than -8 on the number line.
q + 1/3 > 1/2 (1 point)
Choose the correct solution and graph for the inequality.
To solve the inequality q + 1/3 > 1/2, you can follow these steps:
Step 1: Subtract 1/3 from both sides of the inequality to isolate the variable q:
q + 1/3 - 1/3 > 1/2 - 1/3
q > 1/2 - 1/3
Step 2: Subtract the fractions 1/2 and 1/3:
1/2 - 1/3 = 3/6 - 2/6 = 1/6
So the inequality becomes:
q > 1/6
Step 3: Graph the solution on a number line. Draw an open circle on 1/6 and shade the line to the right, indicating that q is greater than 1/6.
The correct solution is q > 1/6, and the graph would be represented as:
--------o======>
Choose the correct solution and graph for the inequality.
z + 9 <= 6 (1 point)
To solve the inequality z + 9 <= 6, you can follow these steps:
Step 1: Subtract 9 from both sides of the inequality to isolate the variable z:
z + 9 - 9 <= 6 - 9
z <= -3
Step 2: Graph the solution on a number line. Draw a closed circle on -3 and shade the line to the left, indicating that z is less than or equal to -3.
The correct solution is z <= -3, and the graph would be represented as:
o======<
Choose the correct solution and graph for the inequality.
y/3 >= - 5
(1point)
To solve the inequality y/3 >= -5, you can follow these steps:
Step 1: Multiply both sides of the inequality by 3 to isolate the variable y:
3*(y/3) >= -5*3
y >= -15
Step 2: Graph the solution on a number line. Draw a closed circle on -15 and shade the line to the right, indicating that y is greater than or equal to -15.
The correct solution is y >= -15, and the graph would be represented as:
------o======>
Choose the correct solution and graph for the inequality. - y/2 <= 3
(1point)
To solve the inequality -y/2 <= 3, you can follow these steps:
Step 1: Multiply both sides of the inequality by -2 to isolate the variable y:
-2 * (-y/2) >= 3 * (-2)
y >= -6
Step 2: Graph the solution on a number line. Draw a closed circle on -6 and shade the line to the right, indicating that y is greater than or equal to -6.
The correct solution is y >= -6, and the graph would be represented as:
------o======>
Choose the correct solution and graph for the inequality.
2x - 2 < - 12 or 2x + 3 > 7 (1 point)
To solve the inequality 2x - 2 < -12 or 2x + 3 > 7, you can solve each inequality separately and then combine the solutions.
For the first inequality, 2x - 2 < -12:
Step 1: Add 2 to both sides of the inequality to isolate the variable x:
2x - 2 + 2 < -12 + 2
2x < -10
Step 2: Divide both sides of the inequality by 2:
(2x)/2 < (-10)/2
x < -5
For the second inequality, 2x + 3 > 7:
Step 1: Subtract 3 from both sides of the inequality to isolate the variable x:
2x + 3 - 3 > 7 - 3
2x > 4
Step 2: Divide both sides of the inequality by 2:
(2x)/2 > (4)/2
x > 2
To combine the solutions, we can represent it as:
x < -5 or x > 2
To graph the solution, draw an open circle at -5 on the number line and shade to the left, indicating that x is less than -5. Then, draw an open circle at 2 on the number line and shade to the right, indicating that x is greater than 2.
The correct solution is x < -5 or x > 2, and the graph would be represented as:
--------o======================o-------
Choose the correct solution and graph for the inequality.
(3y - 6)/3 + 4 <= - 2or * (4y - 8)/2 - 1 >= 5
(1 point)
To solve the inequality
(3y - 6)/3 + 4 <= -2 or (4y - 8)/2 - 1 >= 5
First, let's simplify each inequality separately:
For the first inequality,
(3y - 6)/3 + 4 <= -2:
Step 1: Distribute the division by 3:
(3y - 6)/3 + 4 <= -2
y - 2 + 4 <= -2
Step 2: Combine like terms:
y + 2 <= -2
For the second inequality,
(4y - 8)/2 - 1 >= 5:
Step 1: Distribute the division by 2:
(4y - 8)/2 - 1 >= 5
2y - 4 - 1 >= 5
2y - 5 >= 5
Now, let's solve each inequality separately:
For y + 2 <= -2:
Step 1: Subtract 2 from both sides of the inequality:
y + 2 - 2 <= -2 - 2
y <= -4
For 2y - 5 >= 5:
Step 1: Add 5 to both sides of the inequality:
2y - 5 + 5 >= 5 + 5
2y >= 10
Step 2: Divide both sides of the inequality by 2:
(2y)/2 >= 10/2
y >= 5
To combine the solutions, we can represent it as:
y <= -4 or y >= 5
To graph the solution, we draw a closed circle at -4 and shade to the left, indicating that y is less than or equal to -4. Then, we draw a closed circle at 5 and shade to the right, indicating that y is greater than or equal to 5.
The correct solution is y <= -4 or y >= 5, and the graph would be represented as:
------o=========================o------
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Your class hopes to collect at least 325 cans of food for the annual food drive. There were 135 cans donated the first week and 89 more the second week.
a. Write an inequality that describes this situation. Let c represent the number of cans of food that must be collected by the end of the third week for your class to meet or surpass its goal. b. How many cans are needed to meet or surpass the goal?
a. To write an inequality that describes the situation, we can use the variable c to represent the number of cans of food that must be collected by the end of the third week for your class to meet or surpass its goal.
The number of cans of food collected by the end of the first week is 135, and the number of cans collected by the end of the second week is 89 more than that. Therefore, the total number of cans collected by the end of the second week is 135 + 89 = 224.
Since your class hopes to collect at least 325 cans of food, we can write the inequality as:
c >= 325
b. To determine how many cans are needed to meet or surpass the goal, we need to subtract the total number of cans already donated by the end of the second week (224) from the goal (325):
325 - 224 = 101
Therefore, your class needs at least 101 more cans to meet or surpass its goal.
Solve.
q + 12 - 2(q - 22) > 0
(2 points)
To solve the inequality q + 12 - 2(q - 22) > 0, you can follow these steps:
Step 1: Distribute the -2 to both terms in the parentheses:
q + 12 - 2q + 44 > 0
Step 2: Combine like terms:
-q + 56 > 0
Step 3: Subtract 56 from both sides of the inequality:
-q + 56 - 56 > 0 - 56
-q > -56
Step 4: Multiply both sides of the inequality by -1 (remembering to reverse the inequality when multiplying or dividing by a negative number):
-q * (-1) < -56 * (-1)
q < 56
The solution to the inequality is q < 56.
Solve.
4x + 6 < - 6
To solve the inequality 4x + 6 < -6, you can follow these steps:
Step 1: Subtract 6 from both sides of the inequality to isolate the variable x:
4x + 6 - 6 < -6 - 6
4x < -12
Step 2: Divide both sides of the inequality by 4 to solve for x:
(4x)/4 < (-12)/4
x < -3
The solution to the inequality is x < -3.
Solve.
- 2/5 * x - 9 < 9/10
(2 points)
To solve the inequality -2/5 * x - 9 < 9/10, you can follow these steps:
Step 1: Add 9 to both sides of the inequality to isolate the variable x:
-2/5 * x - 9 + 9 < 9/10 + 9
-2/5 * x < 99/10
Step 2: Divide both sides of the inequality by -2/5 (remembering to reverse the inequality when multiplying or dividing by a negative number):
(-2/5 * x) / (-2/5) > (99/10) / (-2/5)
x > (99/10) / (-2/5)
Step 3: Simplify the division by multiplying the numerator by the reciprocal of the denominator:
x > (99/10) * (-5/2)
Step 4: Multiply the numerators and denominators:
x > (99 * -5) / (10 * 2)
Step 5: Simplify the numerator and denominator:
x > -495/20
Step 6: Reduce the fraction:
x > -99/4
The solution to the inequality is x > -99/4.