Examples of inequality in high school

Example 4 Solving the Problem

(2) Find the non-negative integer solution of inequality 10(x+4)+x≤84.

Analysis: For the (1) small question, it is necessary to understand that "not less than" means "greater than or equal to", and the symbol means "≥"; (2) A small problem is a non-negative integer, that is, an integer in positive number or zero, so the inequality solution of this problem must be a positive integer or zero. Attention should be paid to the correct application of inequality properties in the process of solving problems.

Solution:

∴ 120-8x≥84-3(4x+ 1)

(2)∵ 10(x+4)+x≤84

∴ 10x+40+x≤84

∴ 1 1x≤44

∴x≤4

Because there are five nonnegative integers not greater than 4, namely 0, 1, 2, 3, 4, the nonnegative integer solution of inequality 10(x+4)+x≤84 is 4, 3, 2, 1 0.

Example 5 Solving Inequalities about X

( 1)ax+2≤bx- 1(2)m(m-x)> n(n-x)

Analysis: The methods and steps to solve the letter coefficient inequality are similar to those to solve the number coefficient inequality, but the letter coefficient is often discussed in the process of solving, which increases the difficulty of the topic. This kind of problem mainly examines the ability to analyze and classify the problem: not only do you need to know when to discuss the classification, but you also need to be able to discuss the classification accurately (which will be explained with the solution of an example).

Solution: (1)∵ax+2≤bx- 1

∴ax-bx≤- 1-2

That is, (a-b)x≤-3

At this time, the form of inequality solution should be obtained according to the different values of X letter coefficient.

That is, (n-m) x > N2-m2

When m > n and n-m < 0, ∴ x < n+m;

When m < n, n-m > 0, ∴ x > n+m;

When m=n, n-m=0, n2=m2, n2-m2=0, the original inequality has no solution. This is because the values on both sides of the inequality are zero and can only be equal, so the inequality does not hold.

Example 6 Solving Inequalities about X

3(a+ 1)x+3a≥2ax+3。

Analysis: Because X is an unknown number, A is regarded as a known number, and A can be any rational number, so when applying the same solution principle, we should distinguish the situations and deal with them separately.

Solution: Remove the brackets and get

3ax+3x+3a≥2ax+3

Move items, get

3ax+3x-2ax≥3-3a

Merge similar projects to obtain

(a+3)x≥3-3a

(3) When a+3=0, that is, a=-3, 0 x ≥ 12 is obtained.

There is no solution to this inequality.

Note: When dealing with letter coefficient inequality, we must first find out which letter is unknown and regard other letters as known numbers. When using the same solution principle to change the unknown coefficient into 1, we should make reasonable classification and discuss them one by one.

Example 7 When m is what value, the solution of equation 3 (2x-3m)-2 (x+4m) = 4 (5-x) about x is not positive.

Analysis: according to the meaning of the question, first solve the equation by taking m as a known number, then list the inequality about m according to the conditions of the solution, and then solve this inequality to find the value or range of m. Note that "non-positive number" is a number less than or equal to zero.

Solution: The known equation is 6x-9m-2x-8m=20-4x.

The solution can be 8x=20+ 17m.

It is known that the solution of the equation is non-positive, so

Example 8 If the solution of equation 5x-(4k- 1)=7x+4k-3 about x is: (1) non-negative and (2) negative, try to determine the value range of k. 。

Analysis: To determine the value range of k, we should regard k as a known number, and work out the solution X of the equation according to the steps of solving a linear equation with one variable (expressed by the algebraic expression of k). At this time, according to whether the solution of the equation is negative or not, we can get the inequality about K, and then we can get the range of K. What needs to be emphasized here is that this problem does not directly solve inequalities, but obtains inequalities according to known conditions, which belongs to the application of inequalities.

Solution: The known equation is 5x-4k+ 1=7x+4k-3.

Solvable -2x=8k-4.

That is, x=2( 1-2k)

(1) It is known that the solution of the equation is non-negative, so

(2) It is known that the solution of the equation is negative, so

Example 9 When the value of x is in what range, the algebraic expression -3x+5:

(1) is a negative number (2) is greater than -4.

(3) Less than -2x+3 (4) Not more than 4x-9

Analysis: The key to solving the problem is to accurately translate written languages such as "negative number", "greater than", "less than" and "not greater than" into digital symbols.

Solution: (1) According to the meaning of the question, it should be inequality.

Solution set of -3x+5 < 0

In order to solve this inequality, you must

(2) According to the meaning of the question, the inequality should be found.

Solution set of -3x+5 >-4

In order to solve this inequality, you must

x 0。

It is complicated to remove brackets directly. Note that all terms on the left contain the factor x-3, which can be solved quickly by using the inverse distribution law.

Solve the original inequality into

(x-3)(278-35 1×2+463)>0，

That is, 39 (x-3) > 0, so x > 3.

8. Clever use of overall merger

Example 9 Solving Inequalities

3 { 2x- 1-[3(2x- 1)+3]} > 5。

Take 2x- 1 as a whole, remove the brackets to get 3 (2x-1)-9 (2x-1)-9 > 5, and combine the whole to get -6 (2x- 1) > 14.

9. Clever disassembly

Example 10 Solving Inequalities

It is considered that dividing -3 into three negative 1 and then combining with the other three items can skillfully solve this problem.

Solving the original inequality is transformed into

X- 1≥0, so X ≥ 1.