Question

Solve the equation. Graph and check your solutions.

|x| + 7 = 11

Answer 1

Solve the equation. Graph and check your solutions.

-3 |x| = -21

Answer 2

Solve the equation.


9 = |m + 5|

Answer 3

Solve the equation.

4 |x - 3| = 32

Answer 4

Solve the absolute value equation.


|5n - 6| + 19 = 3

Answer 5

Solve the equation.


3 |5y - 4| - 3 = 6

Answer 6

Solve and graph the inequality.


|x + 2| => 9

Answer 7

Solve for x.

|4x - 4| => 4

Answer 8

Solve the equation.


|5x| - 8 = 12

Answer 9

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?

Answer 10

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.

Answer 11

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.

Answer 12

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.

Answer 13

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.

Answer 14

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.

Answer 15

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.

Answer 16

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.

Answer 17

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.

Answer 18

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.

Answer 19

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.

Answer 20

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,