There are 4 main types of absolute value equations regarding whether there are;
Type 1: One Absolute Value and a Constant
Solve \( | x-2 | = 5 \).
\( \begin{aligned} \displaystyle
x-2 = 5 &\text{ or } x-2 = -5 \\
\therefore x = 7 &\text{ or } x = -3
\end{aligned} \)
Type 2: One Absolute Value and a Linear Expression
Solve \( | x-1 | = 2x+4 \).
\( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}
(|x-1|)^2 &= (2x+4)^2 &\color{red} \text{squaring both sides} \\
(x-1)^2 &= (2x+4)^2 \\
x^2-2x + 1 &= 4x^2 + 16x + 16 \\
3x^2 + 18x + 15 &= 0 \\
x^2 + 6x + 5 &= 0 \\
(x+5)(x+1) &= 0 \\
x = -5 &\text{ or } x = -1 \\
\color{red} \text{Test } x = -5 \\
\text{LHS} &= |-5-1| \\
&= |-6| \\
&= 6 \\
\text{RHS} &= 2 \times -5 +4 \\
&= -6 \\
\text{LHS} &\ne \text{RHS} \\
\therefore x &\ne -5 \\
\color{red} \text{Test } x = -1 \\
\text{LHS} &= |-1-1| \\
&= |-2| \\
&= 2 \\
\text{RHS} &= 2 \times -1 +4 \\
&= 2 \\
\text{LHS} &= \text{RHS} \\
\therefore x &= -1
\end{aligned} \)
Type 3: Two Absolute Value Expressions
Solve \( | x-1 | = |3-x| \).
\( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}
x-1 &= \pm(3-x) \\
x-1 &= 3-x \color{red} \cdots (1) \\
2x &= 4 \\
x &= 2 \\
x-1 &= -3+x \color{red} \cdots (2) \\
x-x &= -2 \\
0 &= -2 &\color{red} \text{no solution} \\
\therefore x &= 2 &\color{red} \text{from (a) and (2)}
\end{aligned} \)
Type 4: Two Absolute Value Expressions and a Constant
Solve \( | x+2 | + |x-3| = 7 \).
\( \begin{aligned} \displaystyle \require{AMSsymbols}\require{color}
-(x + 2)-(x-3) &= 7 &\color{red} \text{for } x \lt -2 \\
-x-2-x+3 &= 7 \\
-2x &= 6 \\
x &= -3 &\color{red} \text{this is OK for } x \lt -2 \\
(x + 2)-(x-3) &= 7 &\color{red} \text{for } -2 \le x \lt 3 \\
x+2-x+3 &= 7 \\
5 &\ne 7 &\color{red} \text{ no solution}\\
(x + 2) + (x-3) &= 7 &\color{red} \text{for } x \ge 3 \\
x+2+x-3 &= 7 \\
2x &= 8 \\
x &= 4 &\color{red} \text{this is OK for } x \ge 3 \\
\therefore x &= -3 \text{ or } 4
\end{aligned} \)
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FAQs
4 Important Types of Absolute Value Equations – iitutor? ›
Absolute value has the following fundamental properties: Non-negativity |a| ≥ 0. Positive-definiteness |a| = 0a = 0. Multiplicativity |ab| = |a| |b|
What are the 4 properties of absolute value? ›Absolute value has the following fundamental properties: Non-negativity |a| ≥ 0. Positive-definiteness |a| = 0a = 0. Multiplicativity |ab| = |a| |b|
What are the 4 steps to solving an absolute value equation? ›| Step 1: Isolate the absolute value | |3x - 6| - 9 = -3 |3x - 6| = 6 |
|---|---|
| Step 2: Is the number on the other side of the equation negative? | No, it's a positive number, 6, so continue on to step 3 |
| Step 3: Write two equations without absolute value bars | 3x - 6 = 6 |
| Step 4: Solve both equations | 3x - 6 = 6 3x = 12 x = 4 |
More generally, the form of the equation for an absolute value function is y=a| x−h |+k.
What are the different types of absolute value inequalities? ›For example, the expression |x + 3| > 1 is an absolute value inequality containing a greater than symbol. There are four different inequality symbols to choose from. These are less than (<), greater than (>), less than or equal (≤), and greater than or equal (≥).
What are the 4 properties of math explained? ›There are four number properties: commutative property, associative property, distributive property and identity property. Number properties are only associated with algebraic operations that are addition, subtraction, multiplication and division.
What are the 4 number properties? ›There are four basic properties of numbers: commutative, associative, distributive, and identity. You should be familiar with each of these. It is especially important to understand these properties once you reach advanced math such as algebra and calculus.
How do you represent the absolute value of 4? ›The absolute value of 4 is also 4, because 4 is 4 units to the right of 0. Opposites always have the same absolute value because they both have the same distance from 0. The distance from 0 to itself is 0, so the absolute value of 0 is 0.
What is 5 in absolute value equations? ›For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line. Furthermore, the absolute value of the difference of two real numbers is the distance between them.
What are the special cases of absolute value equations? ›When solving equations that involve absolute values, there are two cases to consider. Case 1: The expression inside the absolute value bars is positive. Case 2: The expression inside the absolute value bars is negative. For example, consider the expression | 4x+2 |=8 .
What are the 4 basic forms for an inequality? ›
The four basic inequalities are: less than, greater than, less than or equal to, and greater than or equal to.
What are the 4 properties of inequality? ›- Additive property of inequality:
- Subtraction property of inequality:
- Multiplication property of inequality:
- Division property of inequality:
There are two types of absolute valuation models: Dividend Discount Model and Discounted Cash Flow Model. The absolute valuation method is reductive as the analysis is focused on the characteristics of the company in isolation.
What are the 4 types of math? ›The main branches of mathematics are algebra, number theory, geometry and arithmetic.
What are the four main properties of addition? ›The 4 main properties of addition are commutative, associative, distributive, and additive identity. Commutative refers that the result obtained from addition is still the same if the order changes.
What are the types of properties? ›- Movable property and Immovable property.
- Tangible property and Intangible property.
- Private property and Public property.
- Personal property and Real property.
- Corporeal property Incorporeal property.
The three most common real estate property types are residential, commercial, and land.
What property is 4 * 7 equals 4 * 3 4 * 4? ›First, we can see that the equation is showing a multiplication operation being broken down into two smaller multiplication operations. This is an example of the Distributive Property, which states that a(b + c) = ab + ac. In this case, 4 is being distributed across the sum of 3 and 4, resulting in (4x3) + (4x4).
Why do 4 and 4 have the same absolute value? ›The absolute value of a number is its distance from 0. 4 4 4 and −4 are the same distance from 0, so they have the same absolute value of 4start color #11accd, 4, end color #11accd.
What is the opposite of 4 absolute value? ›The opposite of 4 is -4 (the sign changes from + to -). The opposite of -8 is 8 (the sign changes from - to +).
How do you write an absolute value formula? ›
The most common way to represent the absolute value of a number or expression is to surround it with the absolute value symbol: two vertical straight lines. |6| = 6 means “the absolute value of 6 is 6.” |–6| = 6 means “the absolute value of –6 is 6.”
How many solutions does this absolute value equation have? ›If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. An absolute value equation may have one solution, two solutions, or no solutions.
What is the absolute value of 3? ›For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.
Do all absolute value equations have 2 answers? ›In conclusion, an absolute-value problem will not always have two solutions, because absolute-value inequalities result in one set of solutions. If the problem contains only one absolute number, it will have only one solution and that will be the positive number.
What are the 4 steps in solving quadratic inequalities? ›Identify values of a, b and c to substitute into the quadratic formula. Substitute the values into the quadratic formula. Simplify to calculate the solutions of the inequality. Write the solution using inequality notation.
What are the 4 inequality symbols and what do they mean? ›These inequality symbols are: less than (<), greater than (>), less than or equal (≤), greater than or equal (≥) and the not equal symbol (≠). Inequalities are used to compare numbers and determine the range or ranges of values that satisfy the conditions of a given variable.
What are the 4 ways to write a solution to an inequality? ›There are four ways to represent an inequality: Equation notation, set notation, interval notation, and solution graph.
What are the 4 dimensions of equality? ›Dimensions of Equality: Political Equality, Social Equality and Economic Equality, Danger of Inequalities, Ways to Promote Equality etc.
What are the 3 different types of inequality? ›Related concepts are lifetime Inequality (inequality in incomes for an individual over his or her lifetime), Inequality of Wealth (distribution of wealth across households or individuals at a moment in time), and Inequality of Opportunity (impact on income of circumstances over which individuals have no control, such ...
What are the 4 triangle inequality theorems? ›| S.No | Statement | Reason |
|---|---|---|
| 1. | |CD|= |AC| + |AD| | From figure 3 |
| 2. | |CD|= |AC| + |AB| | AB = AD, ∆ADB is an isosceles triangle |
| 3. | ∠DBA <∠DBC | Since ∠DBC = ∠DBA+∠ABD |
| 4. | ∠ADB<∠DBC | ∆ADB is an isosceles triangle and ∠ADB = ∠DBA |
What are the four types of valuation? ›
When someone refers to four valuation methods, usually they are referring to a discounted cash flow, trading comparables, precedent transactions, and a leverage buyout analysis.
What are the 5 types of valuation? ›This course examines in detail the five key property valuation methods: comparison, investment, residual, profits, and cost-based.
What is the absolute value model? ›The absolute valuation formula is a way to evaluate the value of stocks. You can estimate the absolute value by calculating the company's future cash flow minus capital spending. The company's stock theoretically aligns with its future earning potential.
What are the properties of the absolute value function? ›The absolute value function is used to measure the distance between two numbers. Thus, the distance between x and 0 is |x − 0| = |x|, and the distance between x and y is |x − y|. Thus, the distance from −2 to −4 is |−2 − (−4)| = |−2+4| = |2| = 2, and the distance from −2 to 5 is |−2 − 5| = |−7| = 7. |xy| = |x| |y| .
What are absolute properties? ›Key Takeaways. An absolute title is a property title that gives an unequivocal right of ownership to the owner and any buyers to whom the property is sold. Absolute titles have no liens, attachments, or encumbrances to them. The holder of the absolute title is free to sell the property at their discretion.
What are the rules for absolute value inequalities? ›To solve an absolute value inequality involving “less than,” such as |X|≤p, replace it with the compound inequality −p≤X≤p and then solve as usual. To solve an absolute value inequality involving “greater than,” such as |X|≥p, replace it with the compound inequality X≤−p or X≥p and then solve as usual.
What is the property of absolute value inequality? ›The property of absolute value tells us that ∣ a ∣ = a |a| = a ∣a∣=a for non-negative a a a, so in this case ∣ x + 3 ∣ < 7 ⟹ x + 3 < 7 ⟹ x < 4.
What is an example of an absolute value function? ›To solve an equation such as 8=|2x−6|, we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently. Knowing how to solve problems involving absolute value functions is useful.
What are the properties of absolute value matrix? ›Absolute Value of matrices:
Absolute value of a matrix is nothing but the determinant of that matrix. The determinant can be calculated only for a square matrix. If a square matrix A = [ a 11 a 12 a 21 a 22 ] then its determinant is given by | A | = ( a 11 × a 22 ) − ( a 21 × a 12 ) .
The double absolute value sign or generally represents what is called the norm of the given vector. Another way to think of the double absolute value sign is as the geometric length or magnitude of vector .
What is the value 4? ›
Here the digit 4 is in the tens column. Hence, the place value of the digit 4 is tens or 10s.
How do you find absolute value? ›The absolute value of the number is defined as its distance from the origin. For example, to find the absolute value of 7, locate 7 on the real line and then find its distance from the origin. To find the absolute value of −7, locate −7 on the real line and then find its distance from the origin.