# Solve the absolute inequalityand report your answer using interval notation. |2 – 2x| – 3 < 5

Question

Solve the absolute inequalityand report your answer using interval notation. |2 – 2x| – 3 < 5

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2 years 2021-08-03T09:55:37+00:00 1 Answers 23 views 0

Step-by-step explanation:1. An identity is true for all values of the literal and arithmetical numbers in it.

Example 1 5 x 4 = 20 is an identity.

Example 2 2 + 3 = 5 is an identity.

Example 3 2x + 3x = 5x is an identity since any value substituted for x will yield an equality.

Then

Example 4 x + 3 = 9 is true only if the literal number x = 6.

Example 5 3x – 4 = 11 is true only if x = 5.

Example 6 Solve for x: x + 3 = 7

Solution

To have a true statement we need a value for x that, when added to 3, will yield 7. Our knowledge of arithmetic indicates that 4 is the needed value. Therefore the solution to the equation is x = 4.

What number added to 3 equals 7?

Example 7 Solve for x: x – 5 = 3

Solution

x^2+5x+6=0

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Equations can be classified in two main types:

1. An identity is true for all values of the literal and arithmetical numbers in it.

Example 1 5 x 4 = 20 is an identity.

Example 2 2 + 3 = 5 is an identity.

Example 3 2x + 3x = 5x is an identity since any value substituted for x will yield an equality.

2. A conditional equation is true for only certain values of the literal numbers in it.

Example 4 x + 3 = 9 is true only if the literal number x = 6.

Example 5 3x – 4 = 11 is true only if x = 5.

The literal numbers in an equation are sometimes referred to as variables.

Finding the values that make a conditional equation true is one of the main objectives of this text.

A solution or root of an equation is the value of the variable or variables that make the equation a true statement.

The solution or root is said to satisfy the equation.

Solving an equation means finding the solution or root

Example 6 Solve for x: x + 3 = 7

Solution

To have a true statement we need a value for x that, when added to 3, will yield 7. Our knowledge of arithmetic indicates that 4 is the needed value. Therefore the solution to the equation is x = 4.

What number added to 3 equals 7?

Example 7 Solve for x: x – 5 = 3

Solution

What number do we subtract 5 from to obtain 3? Again our experience with arithmetic tells us that 8 – 5 = 3. Therefore the solution is x = 8.

Example 8 Solve for x: 3x = 15

Solution

What number must be multiplied by 3 to obtain 15? Our answer is x = 5.

Solution

What number do we divide 2 by to obtain 7? Our answer is 14.

Example 10 Solve for x: 2x – 1 = 5

Solution

We would subtract 1 from 6 to obtain 5. Thus 2x = 6. Then x = 3.

Regardless of how an equation is solved, the solution should always be checked for correctness.

Example 11 A student solved the equation 5x – 3 = 4x + 2 and found an answer of x = 6. Was this right or wrong?

Solution

Does x = 6 satisfy the equation 5x – 3 = 4x + 2? To check we substitute 6 for x in the equation to see if we obtain a true statement.

This is not a true statement, so the answer x = 6 is wrong.

Another student solved the same equation and found x = 5.

This is a true statement, so x = 5 is correct.

Many students think that when they have found the solution to an equation, the problem is finished. Not so! The final step should always be to check the solution.

Not all equations can be solved mentally. We now wish to introduce an idea that is a step toward an orderly process for solving equations.

Is x = 3 a solution of x – 1 = 2?

Is x = 3 a solution of 2x + I = 7?

What can be said about the equations x – 1 = 2 and 2x + 1 = 7?

Two equations are equivalent if they have the same solution or solutions

Example 12 3x = 6 and 2x + 1 = 5 are equivalent because in both cases x = 2 is a solution.

Techniques for solving equations will involve processes for changing an equation to an equivalent equation. If a complicated equation such as 2x – 4 + 3x = 7x + 2 – 4x can be changed to a simple equation x = 3, and the equation x = 3 is equivalent to the original equation, then we have solved the equation.

Two questions now become very important.

Are two equations equivalent?

How can we change an equation to another equation that is equivalent to it?

The answer to the first question is found by using the substitution principle.

Example 13 Are 5x + 2 = 6x – 1 and x = 3 equivalent equations?