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Chapter 2: Problem 96

Solve using the addition and multiplication principles. $$ \frac{4}{5}(3 x+4) \leq 20 $$

Short Answer

Expert verified
x \ 7

Step by step solution

01

- Eliminate the Fraction

Multiply both sides of the inequality by 5 to eliminate the fraction. \[ \frac{4}{5}(3x + 4) \times 5 \ \ \text{and} \ 20 \times 5 \] This simplifies to: \[ 4(3x + 4) \ \ 100 \]
02

- Distribute the 4

Distribute the 4 on the left-hand side: \[ 4 \times 3x + 4 \times 4 \ \ 12x + 16 \] Inequality now reads: \[ 12x + 16 \ \ \text{and} \ 100 \]
03

- Move the Constant Term

Subtract 16 from both sides to isolate the term with the variable on one side: \[ 12x + 16 - 16 \ 100 - 16 \] This simplifies to: \[ 12x \ \ 84 \]
04

- Divide by the Coefficient of x

Divide both sides by 12 to solve for x: \[ \frac{12x}{12} \ \frac{84}{12} \] This simplifies to: \[ x \ 7 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

addition principle
The addition principle is a fundamental concept in solving inequalities and equations. It means that you can add or subtract the same value from both sides of the inequality without changing its direction. For example, if you have the inequality \( a \leq b \), adding a number \( c \) to both sides results in \( a + c \leq b + c \). This helps in moving constants to one side to isolate the variable.
In the step-by-step solution, subtracting 16 from both sides illustrates this principle, allowing the term with the variable to stand alone.
multiplication principle
The multiplication principle is another key concept in solving inequalities. It states that you can multiply or divide both sides of an inequality by the same nonzero number without altering its direction. For instance, multiplying both sides of \( a \leq b \) by a positive number \( c \) results in \( ac \leq bc \).
In the exercise, multiplying both sides by 5 to eliminate the fraction demonstrates this principle. Remember, if the number is negative, you must reverse the inequality sign.
distributive property
The distributive property is used to multiply through grouped terms. It says that \( a(b + c) = ab + ac \). This property helps simplify expressions by distributing the multiplication over addition or subtraction.
In the solution, the distribution of 4 across \( 3x + 4 \) results in \( 12x + 16 \). This step ensures we handle each term properly to simplify the process of isolating the variable.
isolating variables
Isolating variables is critical in solving equations and inequalities. It means getting the variable by itself on one side of the inequality to find its value. This often involves using the addition and multiplication principles along with the distributive property.
In the given exercise, isolating the variable \( x \) involves subtracting 16 from both sides and then dividing by 12. Each step systematically reduces the equation until \( x \) is by itself, allowing us to find its value.

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