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Chapter 1: Problem 15

Mentally solve each equation. $$ 2 x-3=0 $$

Short Answer

Expert verified
The solution is \(x = 1.5\)

Step by step solution

01

- Isolate the variable term

To isolate the variable term, add 3 to both sides of the equation:\[2x - 3 + 3 = 0 + 3\]This simplifies to:\[2x = 3\]
02

- Solve for the variable

Divide both sides of the equation by 2 to solve for \(x\):\[\frac{2x}{2} = \frac{3}{2}\]This simplifies to:\[x = \frac{3}{2}\]or equivalently,\[x = 1.5\]

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Key Concepts

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

Isolating the Variable
When solving linear equations, a common goal is isolating the variable. This means we want to get the variable alone on one side of the equation. Consider our example equation:

\[2x - 3 = 0\]

To isolate the term containing the variable (\
Addition Property of Equality
The addition property of equality states that if you add the same number to both sides of an equation, the equality is still true. In our example:

\[2x - 3 = 0\]

To get closer to isolating the variable term (\
Division Property of Equality
The division property of equality means that dividing both sides of an equation by the same non-zero number keeps the equality true. For our simplified equation:

\[2x = 3\]

We divide both sides by 2:

\[\frac{2x}{2} = \frac{3}{2}\]

This results in
\[x = \frac{3}{2}\]

Or as a decimal,
x = 1.5.

This step is crucial because it completes the process of solving for the variable.

Remember, you can always check your work by substituting
the variable back into the original equation to see if the equality holds.

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Most popular questions from this chapter

Dimensions of a Patio A contractor orders 8 cubic yards of premixed cement, all of which is to be used to pour a patio that will be 4 inches thick. If the length of the patio is specified to be twice the width, what will be the patio dimensions? (1 cubic yard = 27 cubic feet)

Reducing the Size of a Candy Bar A jumbo chocolate bar with a rectangular shape measures 12 centimeters in length, 7 centimeters in width, and 3 centimeters in thickness. Due to escalating costs of cocoa, management decides to reduce the volume of the bar by \(10 \%\). To accomplish this reduction, management decides that the new bar should have the same 3 -centimeter thickness, but the length and width of each should be reduced an equal number of centimeters. What should be the dimensions of the new candy bar?

Find the real solutions, if any, of each equation. Use any method. $$ x^{2}-6=0 $$

Find the real solutions of each equation. Use a calculator to express any solutions rounded to two decimal places. $$ x^{4}+\sqrt{3} x^{2}-3=0 $$

Critical Thinking You are the manager of a clothing store and have just purchased 100 dress shirts for $20.00 each. After 1 month of selling the shirts at the regular price, you plan to have a sale giving 40% off the original selling price. However, you still want to make a profit of 4 on each shirt at the sale price. What should you price the shirts at initially to ensure this? If, instead of 40% off at the sale, you give 50% off, by how much is your profit reduced?
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