Absolute value is one of the key contents of the first semester of junior high school, can be combined with a number of knowledge points, is the final exam in the frequency of knowledge points, this article mainly introduces the application of absolute value, divided into seven types of question types classified review.
Question 1: Simplify absolute values
The absolute value of the positive number is itself, the absolute value of the negative number is its opposite number, the absolute value of 0 is 0, if it is a specific number to find its absolute value generally will not go wrong, the key is the topic containing letters, many students are prone to errors.
Analysis: Simplification by using the properties of absolute values, first judging the positive and negative of the algebraic formula in the absolute value according to the value range of the unknown, and then removing the absolute value according to “the absolute value of the positive number is equal to itself, and the absolute value of the negative number is equal to the opposite number”, and the property of the integer addition and subtraction method is used to simplify the evaluation.
Question type 2: Knowing the absolute value of a number to find this number
Similar to question type one, it also uses the basic properties of absolute values for evaluation.
Example 2: known: x>y, and |x|=3,|y|=4, find 2x+y
Analysis: According to the definition of absolute value, from |x|=3, |y|=4, x=±3, y=±4.
This question mainly examines absolute values, size comparison of rational numbers, algebraic evaluation, and proficiency in absolute values, the size relationship of rational numbers, and the mixed operation of rational numbers are the keys to solving this problem.
Example 3: (1) Write out all integers whose absolute value is not greater than 4; (2) Satisfaction (1)
The sum of all integers.
Analysis: (1) According to the concept of absolute value: the distance between a number and the origin on the number line is called the absolute value of the number, and all integers whose absolute value is not greater than 4 have 0, ±, 1, ± 2, ± 3, ± 4; (2) The addition rule of rational numbers can be used.
Solution: (1) All integers with an absolute value not greater than 4 have 0,±1,±2,±3,±4;(2)(-4)+(-3)+(-2)+(-1)+0+1+2+3+4=0.
Type 3: Evaluate the value range
Similar to Type One and Type Two, the value is still evaluated according to the nature of the absolute value, and the algebraic formula in the absolute value itself is non-negative; If you get the opposite number by removing the absolute value, then the algebraic formula in the absolute value is nonpositive.
Example 4: Given |x-2|=2-x, find the value range of x
Analysis: This question examines the absolute value, and understanding that the absolute value of a positive number is equal to itself, the absolute value of a negative number is equal to its opposite number, and the absolute value of 0 is equal to 0 is the key to solving the problem.
Solution: ∵|x-2|=2-x, ∴x-2≤0, solution: x≤2.
If a>0, then |a|=a; If a=0, then |a|=0; if a<0, then |a|=-a. If |a|=a, then a≥0; If |a|=-a, then a≤0.
Type 4: Compare size
If the size of two negative numbers is compared, the size of the absolute value can be compared first, if the absolute value is large, it is small in itself; If the absolute value is small, it is large in itself.
Analysis: first simplify the absolute value, get two negative numbers, compare the size of the two negative numbers, you can first compare the size of the absolute value of the two negative numbers.
On the number axis, the right point represents a larger number than the left point.
Type 5: Non-negativity of absolute values
The sum of several nonnegative algebraic equations is 0, then all algebraic equations are equal to 0.
Example 6: Given |a-2|+|3-b|+|c-4|=0, find the values of the following equations: (1) a+b-c; （2）|-a|+|c|-|-b|．
Analysis: (1) According to the nature of non-negative numbers, “the sum of several non-negative numbers is 0, and the values of these non-negative numbers are 0” to solve the values of a, b, and c, and then substitute them into the calculation; (2) It can be calculated according to the nature of the absolute value.
This question examines the nature of non-negative numbers. The key to solving the problem is to grasp the nature of non-negative numbers: the sum of finite non-negative numbers is zero, then every addition must also be zero.
Type 6: Application of absolute values to the maximum value
Example 6: Answer the following questions based on |x| being a non-negative number and the smallest number of non-negative numbers being 0:
(1) When what value is x, |x-2020|has a minimum value, what is this minimum value?
(2) When what value is x, 2020-|x-1|has a maximum value, what is this maximum value?
(3) Find the minimum value of |x-4|+|x-5|;
(4) Find the minimum value of |x-7|+|x-8|+|x-9|.
Solution: (1) When x=2020, |x-2020| has a minimum value, and this minimum value is 0; (2) When x=1, 2020-|x-1| has a maximum value, and this maximum value is 2020
This question examines the relationship between the distance between two points on the number line and the calculation of the absolute value of the number, and the expression of the algebraic formula after removing the absolute value is the key to solving the problem.
Type 7: Application of absolute values in practice
Example 7: An enterprise produces bottled edible blended oil, according to the quality requirements, the net content (excluding packaging) can have an error of 0.0021 liters, and 6 bottles of edible blended oil are randomly inspected.
How many of these 6 bottles of edible blending oil meet the requirements? Please explain the reason with knowledge of absolute values.
∴ 4 of these 6 bottles of edible blended oil meet the requirements.
This question examines the application of positive and negative numbers in real life, the meaning of absolute values, and the opposite meaning when expressing actual physical quantities with positive and negative numbers, and the quantity of opposite meaning contains two factors: one is the opposite meaning; The second is that they are all quantities, and they are the same kind of quantities.