The absolute value of a number is the distance on a number line between the graph of the number and the origin.
The distance between -3 and the origin is 3. Thus,
|-3| = 3
The distance between 3 and the origin is 3. Thus,
|3| = 3
Therefore, if |x| = 3, then x = 3 or x = -3.
An absolute value function can be written using two or more expressions such as
To solve any absolute value function, it has to be in the form of
|x + a| = k
Here, a and k are real numbers. And there should be only absolute part on the left side.
Let us consider the absolute value equation given below.
|2x + 3| = 5
The following steps will be useful to solve absolute value equations.
Step 1 :
Get rid of absolute sign and split up into two branches.
Step 2 :
For the first branch, take the sign as it is on the right side.
Step 3 :
For the second branch, change the sign on the right side.
Step 4 :
Then solve both the branches.
To sketch the graph of y = a|x + h| + k, use the following steps.
1. Find the x-coordinate of the vertex by finding the value of x for which x + h = 0.
2. Make a table of values using the -xcoordinate of the vertex. Find two -xvalues to its left and two to its right.
3. Plot the points from inside the table. If a > 0, the vertex will be the minimum point and if a < 0, the vertex will be the maximum point.
Table of values :
The graph of y = |x|.
Problems 1-3 : Solve the absolute value equation.
Problem 1 :
|3x - 5| = 7
Solution :
3x - 5 = 7 or 3x - 5 = -7
3x = 12 or 3x = -2
x = 4 or x = -⅔
Problem 2 :
|x + 5| = 0
Solution :
|x + 5| = 0
x + 5 = 0
x = -5
Problem 3 :
|x - 2| = -5
Solution :
|x - 2| = -5 means that the distance between x and 2 is -5. Since distance cannot be negative, the given absolute value equation has no solution.
Problem 4 :
Which of the following expressions is equal to -1 for some values of x?
(A) |1 - x| + 6
(B) |1 - x| + 4
(C) |1 - x| + 2
(D) |1 - x| - 2
Solution :
By definition, the absolute value of any expression is a nonnegative number.
In options (A), (B) and (C),
|1 - x| + 6 > 0
|1 - x| + 4 > 0
|1 - x| + 2 > 0
So, the expressions in options (A), (B) and (C) can not be equal to -1.
In option (D), |1 - x| - 2 could be a negative value.
|1 - x| - 2 = -1
|1 - x| = 1
1 - x = 1 or 1 - x = -1
-x = 0 or -x = -2
x = 0 or x = 2
Therefore, the correct answer is option (D).
Problem 5 :
If |2x + 7|= 5, which of the following could be the value of x?
(A) -6
(B) -4
(C) -2
(D) 0
Solution :
|2x + 7|= 5
2x + 7 = 5 or 2x + 7 = -5
2x = -2 or 2x = -12
x = -1 or x = -6
Therefore, the correct answer is option (A).
Problem 6 :
For what value of x is |x - 1| - 1 equal to 1?
(A) -1
(B) 0
(C) 1
(D) 2
Solution :
|x - 1| - 1 = 1
x - 1 = 1 or x - 1 = -1
x = 2 or x = 0
Therefore, the correct answer is option (B).
Problem 7 :
For what value of x is |3x - 5| equal to -1?
(A) -2
(B) -1
(C) 0
(D) There is no such value of x.
Solution :
Given :
|3x - 5| = -1
Absolute value of any expression can never be negative. So, the equation |3x - 5| = -1 has no solution.
Therefore, the correct answer is option (D).
Problem 8 :
For what value of x is 2|x + 5| - 8 equal to 0?
Solution :
2|x + 5| - 8 = 0
2|x + 5| = 8
|x + 5| = 4
x + 5 = 4 or x + 5 = -4
x = -1 or x = -9
Problem 9 :
For what value of x is 3 - |3 - x| equal to 3?
Solution :
3 - |3 - x| = 3
-|3 - x| = 0
|3 - x| = 0
3 - x = 0
x = 3
Problem 10 :
The graph of the function f is shown in the xy-plane above. For what value of x is the value of f(x) at its maximum?
(A) -3
(B) -1
(C) 1
(D) 3.
Solution :
The maximum value of the function corresponds to the y- coordinate of the point on the graph, which is highest along the vertical axis. The highest point along the y- axis has coordinates (1, 4).
When f(x) is maximum, the value of x is 1.
Therefore, the correct answer is option (C).
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