Solve the inequality. Write the solution in set-builder notation.
−2x−17≥−27
To solve the inequality, we need to isolate the variable x on one side of the inequality.
Starting with −2x − 17 ≥ −27, we can first add 17 to both sides to eliminate the constant term:
−2x − 17 + 17 ≥ −27 + 17
Simplifying, we get:
−2x ≥ −10
Next, divide both sides by −2. Since we are dividing by a negative number, the inequality sign must be reversed:
−2x/−2 ≤ −10/−2
Simplifying further, we obtain:
x ≤ 5
Therefore, the solution to the inequality is x ≤ 5. In set-builder notation, this can be written as:
{x | x ≤ 5}
To solve the inequality −2x − 17 ≥ −27, we need to isolate the variable x.
Step 1: Move the constant term to the other side of the inequality by adding 17 to both sides:
−2x − 17 + 17 ≥ −27 + 17
−2x ≥ −10
Step 2: Divide both sides of the inequality by -2. Remember that when dividing an inequality by a negative number, the inequality sign must be flipped:
(-2x)/-2 ≤ (-10)/-2
x ≤ 5
Therefore, the solution to the inequality is x ≤ 5.
Now let's write the solution in set-builder notation. In set-builder notation, we express the solution as a set of numbers that satisfy the inequality. In this case, the set of numbers that are less than or equal to 5 is written as:
{x | x ≤ 5}
To solve the inequality −2x − 17 ≥ −27, we can start by isolating x.
First, we can add 17 to both sides of the inequality to get:
−2x ≥ −27 + 17
Simplifying, we have:
−2x ≥ −10
Next, we can divide both sides of the inequality by −2. However, remember that when dividing or multiplying both sides of an inequality by a negative number, the direction of the inequality sign should be reversed. So, we get:
x ≤ −10/−2
Simplifying, we have:
x ≤ 5
Therefore, the solution to the inequality is x ≤ 5.
In set-builder notation, we can represent this solution as:
{x | x ≤ 5}
6(2x−3) < 30
To solve the inequality, we will start by distributing the 6 to the terms inside the parentheses:
12x - 18 < 30
Next, we will isolate the variable x by adding 18 to both sides of the inequality:
12x - 18 + 18 < 30 + 18
Simplifying, we get:
12x < 48
Finally, we will divide both sides of the inequality by 12 to solve for x:
12x/12 < 48/12
Simplifying further, we obtain:
x < 4
Therefore, the solution to the inequality is x < 4. In set-builder notation, this can be written as:
{x | x < 4}
Your test scores in one class are 83 and 89. What possible scores can you earn on your next test to have a test average between 86 and 90, inclusive?
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Your score on the third test must be between enter your response here and enter your response here, inclusive.
(Use ascending order.)
To find the possible scores you can earn on your next test, we need to calculate the average of the three test scores and determine the acceptable range for the average.
First, let's calculate the average of the two test scores you already have:
Average = (83 + 89) / 2 = 86
Next, let's determine the lowest and highest possible scores on the third test to achieve an average between 86 and 90, inclusive.
To have an average of 86 or higher, the sum of the three test scores must be at least 3 * 86 = 258.
To have an average of 90 or lower, the sum of the three test scores must be no more than 3 * 90 = 270.
Since you already have a sum of 83 + 89 = 172, subtracting this sum from the lowest and highest possible sums, we get:
Lowest possible score on the third test = 258 - 172 = 86
Highest possible score on the third test = 270 - 172 = 98
Therefore, the possible scores you can earn on your next test to have a test average between 86 and 90, inclusive, are between 86 and 98. In set-builder notation, this can be written as:
{score | 86 ≤ score ≤ 98}
9<4p-3 ≤25
We have the inequality 9 < 4p - 3 ≤ 25 that we want to solve.
First, we can begin by adding 3 to all parts of the inequality:
9 + 3 < 4p - 3 + 3 ≤ 25 + 3
This simplifies to:
12 < 4p ≤ 28
Next, we can divide all parts of the inequality by 4 to solve for p:
12/4 < 4p/4 ≤ 28/4
This simplifies to:
3 < p ≤ 7
Therefore, the solution to the inequality is 3 < p ≤ 7. In set-builder notation, this can be written as:
{p | 3 < p ≤ 7}
The solutions are given by?
The solutions to the inequality 9 < 4p - 3 ≤ 25 are given by the interval (3, 7] in interval notation. In set-builder notation, the solutions can be written as:
{p | 3 < p ≤ 7}
−7<2p−3≤11
To solve the inequality, we will isolate the variable p.
Starting with −7 < 2p - 3 ≤ 11, we can first add 3 to all parts of the inequality:
-7 + 3 < 2p - 3 + 3 ≤ 11 + 3
This simplifies to:
-4 < 2p ≤ 14
Next, we can divide all parts of the inequality by 2 to solve for p:
-4/2 < 2p/2 ≤ 14/2
This simplifies to:
-2 < p ≤ 7
Therefore, the solution to the inequality is -2 < p ≤ 7. In set-builder notation, this can be written as:
{p | -2 < p ≤ 7}
Solve and graph the equation.
x=5
The equation x = 5 represents a vertical line that passes through the point (5, 0) on the x-axis.
To graph this equation, we can plot the point (5, 0) and draw a vertical line passing through that point.
Here is a visual representation of the graph:
```
|
|
|
x______|______
```
The vertical line intersects the x-axis at x = 5 and is parallel to the y-axis.
Solve and graph the equation.
2t=8
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Part 1
Select the correct choice and, if necessary, fill in the answer box in your choice below.
A.
t = enter your response here
(Simplify your answer. Use a comma to separate answers as needed.)
B.
There is no solution.
To solve the equation 2t = 8, we need to isolate the variable t.
Dividing both sides of the equation by 2, we have:
t = 8/2
Simplifying, we get:
t = 4
Therefore, the solution to the equation is t = 4.
As for the graph, this equation represents a vertical line that intersects the y-axis at t = 4 and is parallel to the x-axis.
Solve and graph the equation.
4t=24
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Part 1
Select the correct choice and, if necessary, fill in the answer box in your choice below.
A.
t = enter your response here
(Simplify your answer. Use a comma to separate answers as needed.)
B.
There is no solution.
To solve the equation 4t = 24, we need to isolate the variable t.
Dividing both sides of the equation by 4, we have:
t = 24/4
Simplifying, we get:
t = 6
Therefore, the solution to the equation is t = 6.
As for the graph, this equation represents a vertical line that intersects the y-axis at t = 6 and is parallel to the x-axis.
olve and graph the inequality.
x+2≥9
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Part 1
Choose the correct answer below.
A.x less than or equals minus 9 or x greater than or equals 9
x≤−9 or x≥9
B.x less than or equals minus 11 or x greater than or equals 7
To solve the inequality x + 2 ≥ 9, we can start by isolating the variable x.
Subtracting 2 from both sides of the inequality, we have:
x ≥ 9 - 2
Simplifying, we get:
x ≥ 7
Therefore, the solution to the inequality is x ≥ 7.
As for the graph, this inequality represents a number line that starts at 7 and includes all values greater than or equal to 7.
Graphically, it would look like this:
--------------●------------------->
7