To solve the equation, we need to isolate the absolute value term.
First, we subtract 7 from both sides of the equation:
|x| = 11 - 7
|x| = 4
Next, we can write two separate equations without the absolute value:
x = 4 and x = -4
Now, let's graph the solutions on a number line:
--------------o--------------o--------------
-5 -4 -3 -2 -1 0 1 2 3 4 5
The solutions are x = 4 and x = -4, represented by the points on the number line.
To check our solutions, we substitute them back into the original equation:
For x = 4:
|4| + 7 = 11
4 + 7 = 11
11 = 11 (true)
For x = -4:
|-4| + 7 = 11
4 + 7 = 11
11 = 11 (true)
Both solutions satisfy the original equation, so our solutions are correct.
Solve the equation. Graph and check your solutions.
|x| + 7 = 11
20 answers
Solve the equation. Graph and check your solutions.
-3 |x| = -21
-3 |x| = -21
To solve the equation, we need to isolate the absolute value term.
Step 1: Divide both sides of the equation by -3:
-3 |x| / -3 = -21 / -3
|x| = 7
Step 2: Split the equation into two cases:
Case 1: x ≥ 0
In this case, the absolute value expression |x| simplifies to x. So we rewrite the equation as:
x = 7
Case 2: x < 0
In this case, the absolute value expression |x| simplifies to -x. So we rewrite the equation as:
-x = 7
Step 3: Solve each case separately:
For Case 1:
x = 7
For Case 2:
-x = 7
Multiply both sides by -1 to isolate x:
x = -7
Therefore, the solutions to the equation are x = 7 and x = -7.
To graph the equation, we plot the solutions on a number line. Place closed dots on 7 and -7 to indicate that these points are included in the solution.
-7--------------o--------------7
To check if the solutions are correct, substitute them back into the original equation and see if both sides are equal:
For x = 7:
-3 |7| = -21
-3(7) = -21
-21 = -21 (True)
For x = -7:
-3 |-7| = -21
-3(7) = -21
-21 = -21 (True)
Both solutions satisfy the original equation, so our solutions are correct.
Step 1: Divide both sides of the equation by -3:
-3 |x| / -3 = -21 / -3
|x| = 7
Step 2: Split the equation into two cases:
Case 1: x ≥ 0
In this case, the absolute value expression |x| simplifies to x. So we rewrite the equation as:
x = 7
Case 2: x < 0
In this case, the absolute value expression |x| simplifies to -x. So we rewrite the equation as:
-x = 7
Step 3: Solve each case separately:
For Case 1:
x = 7
For Case 2:
-x = 7
Multiply both sides by -1 to isolate x:
x = -7
Therefore, the solutions to the equation are x = 7 and x = -7.
To graph the equation, we plot the solutions on a number line. Place closed dots on 7 and -7 to indicate that these points are included in the solution.
-7--------------o--------------7
To check if the solutions are correct, substitute them back into the original equation and see if both sides are equal:
For x = 7:
-3 |7| = -21
-3(7) = -21
-21 = -21 (True)
For x = -7:
-3 |-7| = -21
-3(7) = -21
-21 = -21 (True)
Both solutions satisfy the original equation, so our solutions are correct.
Solve the equation.
9 = |m + 5|
9 = |m + 5|
To solve the equation, we need to isolate the absolute value term.
Step 1: We can rewrite the equation as two separate equations:
m + 5 = 9 or m + 5 = -9
Step 2: Solve each equation separately:
For m + 5 = 9:
Subtract 5 from both sides:
m = 9 - 5
m = 4
For m + 5 = -9:
Subtract 5 from both sides:
m = -9 - 5
m = -14
Therefore, the solutions to the equation are m = 4 and m = -14.
To check if the solutions are correct, substitute them back into the original equation and see if both sides are equal:
For m = 4:
9 = |4 + 5|
9 = |9|
9 = 9 (True)
For m = -14:
9 = |-14 + 5|
9 = |-9|
9 = 9 (True)
Both solutions satisfy the original equation, so our solutions are correct.
Step 1: We can rewrite the equation as two separate equations:
m + 5 = 9 or m + 5 = -9
Step 2: Solve each equation separately:
For m + 5 = 9:
Subtract 5 from both sides:
m = 9 - 5
m = 4
For m + 5 = -9:
Subtract 5 from both sides:
m = -9 - 5
m = -14
Therefore, the solutions to the equation are m = 4 and m = -14.
To check if the solutions are correct, substitute them back into the original equation and see if both sides are equal:
For m = 4:
9 = |4 + 5|
9 = |9|
9 = 9 (True)
For m = -14:
9 = |-14 + 5|
9 = |-9|
9 = 9 (True)
Both solutions satisfy the original equation, so our solutions are correct.
Solve the equation.
4 |x - 3| = 32
4 |x - 3| = 32
To solve the equation, we need to isolate the absolute value term.
Step 1: Divide both sides of the equation by 4:
4 |x - 3| / 4 = 32 / 4
|x - 3| = 8
Step 2: Split the equation into two cases:
Case 1: x - 3 ≥ 0
In this case, the absolute value expression |x - 3| simplifies to x - 3. So we rewrite the equation as:
x - 3 = 8
Case 2: x - 3 < 0
In this case, the absolute value expression |x - 3| simplifies to -(x - 3), which is the same as 3 - x. So we rewrite the equation as:
3 - x = 8
Step 3: Solve each case separately:
For Case 1:
x - 3 = 8
Add 3 to both sides:
x = 8 + 3
x = 11
For Case 2:
3 - x = 8
Subtract 3 from both sides:
-x = 8 - 3
-x = 5
Multiply both sides by -1 to isolate x:
x = -5
Therefore, the solutions to the equation are x = 11 and x = -5.
To check if the solutions are correct, substitute them back into the original equation and see if both sides are equal:
For x = 11:
4 |11 - 3| = 32
4 |8| = 32
4(8) = 32
32 = 32 (True)
For x = -5:
4 |-5 - 3| = 32
4 |-8| = 32
4(8) = 32
32 = 32 (True)
Both solutions satisfy the original equation, so our solutions are correct.
Step 1: Divide both sides of the equation by 4:
4 |x - 3| / 4 = 32 / 4
|x - 3| = 8
Step 2: Split the equation into two cases:
Case 1: x - 3 ≥ 0
In this case, the absolute value expression |x - 3| simplifies to x - 3. So we rewrite the equation as:
x - 3 = 8
Case 2: x - 3 < 0
In this case, the absolute value expression |x - 3| simplifies to -(x - 3), which is the same as 3 - x. So we rewrite the equation as:
3 - x = 8
Step 3: Solve each case separately:
For Case 1:
x - 3 = 8
Add 3 to both sides:
x = 8 + 3
x = 11
For Case 2:
3 - x = 8
Subtract 3 from both sides:
-x = 8 - 3
-x = 5
Multiply both sides by -1 to isolate x:
x = -5
Therefore, the solutions to the equation are x = 11 and x = -5.
To check if the solutions are correct, substitute them back into the original equation and see if both sides are equal:
For x = 11:
4 |11 - 3| = 32
4 |8| = 32
4(8) = 32
32 = 32 (True)
For x = -5:
4 |-5 - 3| = 32
4 |-8| = 32
4(8) = 32
32 = 32 (True)
Both solutions satisfy the original equation, so our solutions are correct.
Solve the absolute value equation.
|5n - 6| + 19 = 3
|5n - 6| + 19 = 3
To solve the equation, we need to isolate the absolute value term.
Step 1: Subtract 19 from both sides of the equation:
|5n - 6| + 19 - 19 = 3 - 19
|5n - 6| = -16
At this point, we can see that the absolute value of any number is always non-negative, so it cannot be equal to a negative number. Therefore, the equation has no solutions.
Graphically, the equation corresponds to the absolute value function |5n - 6|, and adding 19 to it shifts the graph upward by 19 units. The line corresponding to y = -16 is a horizontal line below the x-axis with no points in common with the graph of the absolute value function. Therefore, there is no intersection and no solutions to the equation.
In conclusion, the equation |5n - 6| + 19 = 3 has no solutions.
Step 1: Subtract 19 from both sides of the equation:
|5n - 6| + 19 - 19 = 3 - 19
|5n - 6| = -16
At this point, we can see that the absolute value of any number is always non-negative, so it cannot be equal to a negative number. Therefore, the equation has no solutions.
Graphically, the equation corresponds to the absolute value function |5n - 6|, and adding 19 to it shifts the graph upward by 19 units. The line corresponding to y = -16 is a horizontal line below the x-axis with no points in common with the graph of the absolute value function. Therefore, there is no intersection and no solutions to the equation.
In conclusion, the equation |5n - 6| + 19 = 3 has no solutions.
Solve the equation.
3 |5y - 4| - 3 = 6
3 |5y - 4| - 3 = 6
To solve the equation, we need to isolate the absolute value term.
Step 1: Add 3 to both sides of the equation:
3 |5y - 4| - 3 + 3 = 6 + 3
3 |5y - 4| = 9
Step 2: Divide both sides of the equation by 3:
(3 |5y - 4|) / 3 = 9 / 3
|5y - 4| = 3
Step 3: Split the equation into two cases:
Case 1: 5y - 4 ≥ 0
In this case, the absolute value expression |5y - 4| simplifies to 5y - 4. So we rewrite the equation as:
5y - 4 = 3
Case 2: 5y - 4 < 0
In this case, the absolute value expression |5y - 4| simplifies to -(5y - 4), which is the same as 4 - 5y. So we rewrite the equation as:
4 - 5y = 3
Step 4: Solve each case separately:
For Case 1:
5y - 4 = 3
Add 4 to both sides:
5y = 3 + 4
5y = 7
Divide both sides by 5:
y = 7 / 5
y = 1.4
For Case 2:
4 - 5y = 3
Subtract 4 from both sides:
-5y = 3 - 4
-5y = -1
Divide both sides by -5 (note that this changes the direction of the inequality):
y = (-1) / (-5)
y = 1/5
y = 0.2
Therefore, the solutions to the equation are y = 1.4 and y = 0.2.
To check if the solutions are correct, substitute them back into the original equation and see if both sides are equal:
For y = 1.4:
3 |5(1.4) - 4| - 3 = 6
3 |7 - 4| - 3 = 6
3 |3| - 3 = 6
3(3) - 3 = 6
9 - 3 = 6
6 = 6 (True)
For y = 0.2:
3 |5(0.2) - 4| - 3 = 6
3 |1 - 4| - 3 = 6
3 |-3| - 3 = 6
3(3) - 3 = 6
9 - 3 = 6
6 = 6 (True)
Both solutions satisfy the original equation, so our solutions are correct.
Step 1: Add 3 to both sides of the equation:
3 |5y - 4| - 3 + 3 = 6 + 3
3 |5y - 4| = 9
Step 2: Divide both sides of the equation by 3:
(3 |5y - 4|) / 3 = 9 / 3
|5y - 4| = 3
Step 3: Split the equation into two cases:
Case 1: 5y - 4 ≥ 0
In this case, the absolute value expression |5y - 4| simplifies to 5y - 4. So we rewrite the equation as:
5y - 4 = 3
Case 2: 5y - 4 < 0
In this case, the absolute value expression |5y - 4| simplifies to -(5y - 4), which is the same as 4 - 5y. So we rewrite the equation as:
4 - 5y = 3
Step 4: Solve each case separately:
For Case 1:
5y - 4 = 3
Add 4 to both sides:
5y = 3 + 4
5y = 7
Divide both sides by 5:
y = 7 / 5
y = 1.4
For Case 2:
4 - 5y = 3
Subtract 4 from both sides:
-5y = 3 - 4
-5y = -1
Divide both sides by -5 (note that this changes the direction of the inequality):
y = (-1) / (-5)
y = 1/5
y = 0.2
Therefore, the solutions to the equation are y = 1.4 and y = 0.2.
To check if the solutions are correct, substitute them back into the original equation and see if both sides are equal:
For y = 1.4:
3 |5(1.4) - 4| - 3 = 6
3 |7 - 4| - 3 = 6
3 |3| - 3 = 6
3(3) - 3 = 6
9 - 3 = 6
6 = 6 (True)
For y = 0.2:
3 |5(0.2) - 4| - 3 = 6
3 |1 - 4| - 3 = 6
3 |-3| - 3 = 6
3(3) - 3 = 6
9 - 3 = 6
6 = 6 (True)
Both solutions satisfy the original equation, so our solutions are correct.
Solve and graph the inequality.
|x + 2| => 9
|x + 2| => 9
To solve the inequality, we need to consider two cases, one when the absolute value is positive and the other when it is negative.
Case 1: x + 2 ≥ 0
In this case, the absolute value expression |x + 2| simplifies to x + 2. So we can rewrite the inequality as:
x + 2 ≥ 9
Subtracting 2 from both sides, we have:
x ≥ 7
Case 2: x + 2 < 0
In this case, the absolute value expression |x + 2| simplifies to -(x + 2), which is equivalent to -x - 2. So we can rewrite the inequality as:
-x - 2 ≥ 9
Adding 2 to both sides and multiplying by -1 (which reverses the inequality), we obtain:
x ≤ -11
Therefore, the solution to the inequality is -11 ≤ x ≤ 7.
Graphically, this corresponds to a horizontal line segment on the number line extending from -11 to 7 with closed circles indicating that -11 and 7 are included in the solution.
----------------o----------------o-----------------
-11 -2 7
So, to graph the inequality |x + 2| >= 9, we represent the solution on the number line as shown above.
Case 1: x + 2 ≥ 0
In this case, the absolute value expression |x + 2| simplifies to x + 2. So we can rewrite the inequality as:
x + 2 ≥ 9
Subtracting 2 from both sides, we have:
x ≥ 7
Case 2: x + 2 < 0
In this case, the absolute value expression |x + 2| simplifies to -(x + 2), which is equivalent to -x - 2. So we can rewrite the inequality as:
-x - 2 ≥ 9
Adding 2 to both sides and multiplying by -1 (which reverses the inequality), we obtain:
x ≤ -11
Therefore, the solution to the inequality is -11 ≤ x ≤ 7.
Graphically, this corresponds to a horizontal line segment on the number line extending from -11 to 7 with closed circles indicating that -11 and 7 are included in the solution.
----------------o----------------o-----------------
-11 -2 7
So, to graph the inequality |x + 2| >= 9, we represent the solution on the number line as shown above.
Solve for x.
|4x - 4| => 4
|4x - 4| => 4
To solve for x, we need to consider two cases, one when the absolute value expression is positive or zero, and another when it is negative.
Case 1: 4x - 4 ≥ 0
In this case, the absolute value expression |4x - 4| simplifies to 4x - 4. So we can rewrite the inequality as:
4x - 4 ≥ 4
Adding 4 to both sides, we have:
4x ≥ 8
Dividing both sides by 4 (note that this does not reverse the inequality because 4 is positive), we obtain:
x ≥ 2
Case 2: 4x - 4 < 0
In this case, the absolute value expression |4x - 4| simplifies to -(4x - 4), which is equivalent to -4x + 4. So we can rewrite the inequality as:
-4x + 4 ≥ 4
Subtracting 4 from both sides, we have:
-4x ≥ 0
Dividing both sides by -4 (which reverses the inequality because -4 is negative), we obtain:
x ≤ 0
Therefore, the solution to the equation is x ≤ 0 or x ≥ 2.
Graphically, this corresponds to a shaded region on the number line extending from negative infinity to 0, and from 2 to positive infinity, indicating that any value less than or equal to 0 and any value greater than or equal to 2 are solutions to the inequality.
----------------o-------------------o---------------
-∞ 0 2 ∞
So, the solution to the inequality |4x - 4| ≥ 4 is x ≤ 0 or x ≥ 2.
Case 1: 4x - 4 ≥ 0
In this case, the absolute value expression |4x - 4| simplifies to 4x - 4. So we can rewrite the inequality as:
4x - 4 ≥ 4
Adding 4 to both sides, we have:
4x ≥ 8
Dividing both sides by 4 (note that this does not reverse the inequality because 4 is positive), we obtain:
x ≥ 2
Case 2: 4x - 4 < 0
In this case, the absolute value expression |4x - 4| simplifies to -(4x - 4), which is equivalent to -4x + 4. So we can rewrite the inequality as:
-4x + 4 ≥ 4
Subtracting 4 from both sides, we have:
-4x ≥ 0
Dividing both sides by -4 (which reverses the inequality because -4 is negative), we obtain:
x ≤ 0
Therefore, the solution to the equation is x ≤ 0 or x ≥ 2.
Graphically, this corresponds to a shaded region on the number line extending from negative infinity to 0, and from 2 to positive infinity, indicating that any value less than or equal to 0 and any value greater than or equal to 2 are solutions to the inequality.
----------------o-------------------o---------------
-∞ 0 2 ∞
So, the solution to the inequality |4x - 4| ≥ 4 is x ≤ 0 or x ≥ 2.
Solve the equation.
|5x| - 8 = 12
|5x| - 8 = 12
To solve the equation, we need to isolate the absolute value term.
Step 1: Add 8 to both sides of the equation:
|5x| - 8 + 8 = 12 + 8
|5x| = 20
Step 2: Split the equation into two cases:
Case 1: 5x ≥ 0
In this case, the absolute value expression |5x| simplifies to 5x. So we rewrite the equation as:
5x = 20
Case 2: 5x < 0
In this case, the absolute value expression |5x| simplifies to -(5x), which is the same as -5x. So we rewrite the equation as:
-5x = 20
Step 3: Solve each case separately:
For Case 1:
5x = 20
Divide both sides by 5:
x = 20/5
x = 4
For Case 2:
-5x = 20
Divide both sides by -5 (note that this changes the direction of the inequality):
x = 20 / -5
x = -4
Therefore, the solutions to the equation are x = 4 and x = -4.
To check if the solutions are correct, substitute them back into the original equation and see if both sides are equal:
For x = 4:
|5(4)| - 8 = 12
|20| - 8 = 12
20 - 8 = 12
12 = 12 (True)
For x = -4:
|5(-4)| - 8 = 12
|-20| - 8 = 12
20 - 8 = 12
12 = 12 (True)
Both solutions satisfy the original equation, so our solutions are correct.
Step 1: Add 8 to both sides of the equation:
|5x| - 8 + 8 = 12 + 8
|5x| = 20
Step 2: Split the equation into two cases:
Case 1: 5x ≥ 0
In this case, the absolute value expression |5x| simplifies to 5x. So we rewrite the equation as:
5x = 20
Case 2: 5x < 0
In this case, the absolute value expression |5x| simplifies to -(5x), which is the same as -5x. So we rewrite the equation as:
-5x = 20
Step 3: Solve each case separately:
For Case 1:
5x = 20
Divide both sides by 5:
x = 20/5
x = 4
For Case 2:
-5x = 20
Divide both sides by -5 (note that this changes the direction of the inequality):
x = 20 / -5
x = -4
Therefore, the solutions to the equation are x = 4 and x = -4.
To check if the solutions are correct, substitute them back into the original equation and see if both sides are equal:
For x = 4:
|5(4)| - 8 = 12
|20| - 8 = 12
20 - 8 = 12
12 = 12 (True)
For x = -4:
|5(-4)| - 8 = 12
|-20| - 8 = 12
20 - 8 = 12
12 = 12 (True)
Both solutions satisfy the original equation, so our solutions are correct.
Starting from 150 ft away, your friend skates toward you and then passes you. She skates at a constant speed of 20ft/s. Her distance d from you in feet after t seconds is given by d = |150 - 20t|. At what times is she 90 ft from you?
To find the times when your friend is 90 ft away from you, we can set up the equation:
|150 - 20t| = 90
We can split this equation into two cases, one where the expression inside the absolute value is positive and one where it is negative.
Case 1: 150 - 20t ≥ 0
In this case,
|150 - 20t| = 90
We can split this equation into two cases, one where the expression inside the absolute value is positive and one where it is negative.
Case 1: 150 - 20t ≥ 0
In this case,
Starting from 150 ft away, your friend skates toward you and then passes you. She skates at a constant speed of 20ft/s. Her distance d from you in feet after t seconds is given by d = |150 - 20t|. At what times is she 90 ft from you?
She is 90 ft away from you after [ ] s.
She is 90 ft away from you after [ ] s.
2 answers