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Chapter 6: Problem 21

Solve. $$ 4(3 x+5)=0 $$

Short Answer

Expert verified
x = \frac{-5}{3}

Step by step solution

01

Distribute the Constant

Distribute the 4 to both terms inside the parenthesis: \[4(3x + 5) = 4 \times 3x + 4 \times 5\]This simplifies to: \[12x + 20 = 0\]
02

Isolate the Variable Term

Subtract 20 from both sides of the equation to isolate the term with the variable:\[12x + 20 - 20 = 0 - 20\]Which simplifies to:\[12x = -20\]
03

Solve for x

Divide both sides of the equation by 12 to solve for x:\[x = \frac{-20}{12}\]Simplify the fraction:\[x = \frac{-5}{3}\]

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Key Concepts

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

distributive property
The distributive property is a fundamental concept in algebra. It allows you to multiply a single term by each term inside parentheses. This method helps simplify expressions and make them easier to work with. In our exercise, we applied the distributive property to the equation \(4(3x + 5) = 0\).

Here's how it works:
  • Multiply the 4 by each term inside the parentheses: \(4 \times 3x + 4 \times 5\).
  • This simplifies to: \12x + 20 = 0\.
Using the distributive property helps you expand and simplify equations in one clear step.
isolating the variable
Isolating the variable is an essential step in solving linear equations. It means getting the variable (in this case, \(x\)) alone on one side of the equation. This simplifies the equation and makes it easier to solve.

Let's see how we isolated the variable term in our example:
  • We started with: \12x + 20 = 0\.
  • To isolate \(12x\), subtract 20 from both sides: \12x + 20 - 20 = 0 - 20\, which simplifies to: \12x = -20\.
By isolating the variable, we made the next step鈥攕olving for \(x\)鈥攎uch more straightforward.
simplifying fractions
Simplifying fractions is a key skill in algebra. It allows you to rewrite fractions in their simplest form, making them easier to understand and work with.

In our exercise, after isolating the variable \(12x = -20\), we needed to solve for \(x\). Here's how the simplification process works:
  • Divide both sides by 12: \(x = \frac{-20}{12}\).
  • Simplify the fraction: \(x = \frac{-5}{3}\).
Simplifying fractions ensures that your final answers are as clear and concise as possible.

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Most popular questions from this chapter

The formula for density, \(d\), is \(d=\frac{m}{v}\), where \(m\) is the mass and \(v\) is the volume. The density of a steel sphere is \(7.85 \frac{\mathrm{g}}{\mathrm{cm}^{3}}\), and its mass is \(5 \times 10^{2} \mathrm{~g}\). Solve the formula for \(v\), and find the volume of this sphere. Round to the nearest whole number.

Factor completely. Identify any prime polynomials. $$ 27 h^{3}-k^{3} $$

For exercises \(85-102\), factor completely. Identify any prime polynomials. $$ 2 p^{4}+4 p^{2} w^{3}+2 w^{6} $$

A student said that the solutions of \(3(x+8)=0\) are \(x=3\) and \(x=-8\). Explain what is wrong with this thinking.

Solve. $$ 5(x-3)(x-7)=0 $$
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