Solve the equation. Graph and check your solutions.
|x|+1=12
Solve the equation for x. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The given equation is |x| + 1 = 12.
To solve this equation, we need to eliminate the absolute value. There are two cases we need to consider:
Case 1: x is positive or zero
In this case, the equation becomes x + 1 = 12. Solving for x, we have:
x = 12 - 1
x = 11
Case 2: x is negative
In this case, the equation becomes -x + 1 = 12. Solving for x, we have:
-x = 12 - 1
-x = 11
Multiplying both sides by -1 (to isolate x), we get:
x = -11
Therefore, the solutions to the equation are x = 11 and x = -11.
To graphically verify the solutions, we can plot the graph of the equation y = |x| + 1 and check if it intersects the line y = 12.
Here is the graph:
-7 -5 -3 -1 1 3 5 7 9 11 13 15
|----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
7 9 11 13 15 17 19 21 23 25 27 29 31
From the graph, we can see that the graph of y = |x| + 1 intersects the line y = 12 at x = 11 and x = -11, confirming our solutions.
Final answer: The solutions to the equation |x| + 1 = 12 are x = 11 and x = -11.
Solve the equation. Graph and check your solutions.
|x|+8=14
Solve the equation for x. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The given equation is |x| + 8 = 14.
To solve this equation, we need to eliminate the absolute value. There are two cases we need to consider:
Case 1: x is positive or zero
In this case, the equation becomes x + 8 = 14. Solving for x, we have:
x = 14 - 8
x = 6
Case 2: x is negative
In this case, the equation becomes -x + 8 = 14. Solving for x, we have:
-x = 14 - 8
-x = 6
Multiplying both sides by -1 (to isolate x), we get:
x = -6
Therefore, the solutions to the equation are x = 6 and x = -6.
To graphically verify the solutions, we can plot the graph of the equation y = |x| + 8 and check if it intersects the line y = 14.
Here is the graph:
-7 -5 -3 -1 1 3 5 7 9 11 13 15
|----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
1 3 5 7 9 11 13 15 17 19 21 23 25 27
From the graph, we can see that the graph of y = |x| + 8 intersects the line y = 14 at x = 6 and x = -6, confirming our solutions.
Final answer: The solutions to the equation |x| + 8 = 14 are x = 6 and x = -6.
To solve the equation |x| + 1 = 12, we need to isolate the absolute value term and then solve for x. Here's how you can do it step by step:
Step 1: Remove the constant term by subtracting 1 from both sides of the equation:
|x| = 12 - 1
|x| = 11
Step 2: Remove the absolute value brackets. The equation |x| = 11 can be rewritten as two separate equations:
x = 11 or x = -11
Step 3: Graph and check the solutions:
To graphically represent the solutions, plot the points (11,0) and (-11,0) on a number line. Draw an open circle at each point to denote that the equality is not satisfied at those points.
Checking the solutions:
Substitute each value of x back into the original equation |x| + 1 = 12 and see if the equation holds true.
For x = 11:
|11| + 1 = 12
11 + 1 = 12
12 = 12 (True)
For x = -11:
|-11| + 1 = 12
11 + 1 = 12
12 = 12 (True)
Both solutions satisfy the original equation, so x = 11 and x = -11 are valid solutions.
Therefore, the solutions to the equation |x| + 1 = 12 are x = 11 and x = -11.