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Chapter 4: Problem 1

Solve these inequalities. $$ 3 x+5>7 $$

Short Answer

Expert verified
x > \(\frac{2}{3}\)

Step by step solution

01

Isolate the variable term

Subtract 5 from both sides of the inequality to isolate the term with the variable.
02

Simplify the inequality

Perform the subtraction to simplify the inequality: \[3x + 5 - 5 > 7 - 5\] which simplifies to \[3x > 2\]
03

Solve for the variable

Divide both sides by 3 to solve for the variable: \[\frac{3x}{3} > \frac{2}{3}\] which simplifies to \[x > \frac{2}{3}\]

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Key Concepts

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

Isolate the Variable
When solving inequalities, the first step is crucial: isolate the variable. This means you need to get the term with the variable on one side and all constants on the other side.
For this example, we start with the inequality:
\[3x + 5 > 7\]
To move the constant (5) to the other side, we subtract 5 from both sides of the inequality:
\[3x + 5 - 5 > 7 - 5\]
This leaves us with:
\[3x > 2\]Remember, whatever you do to one side of the inequality, you must do to the other. This keeps everything balanced.
Simplify the Inequality
Once the variable is isolated, the next step is to simplify the inequality.
In our example, after isolating the terms, you get:
\[3x > 2\]
Here, '3x' and '2' are already in their simplest form. Simplifying makes the inequality easier to interpret and solve.
Always make sure no unnecessary constants are present with the variable. Simplifying might also involve combining like terms or reducing fractions, but in our case, it鈥檚 straightforward. Double-check to ensure everything is simplified before moving on.
Solve for the Variable
The final step is to solve for the variable by performing different operations. For the inequality:
\[3x > 2\]
We need to solve for 'x'. Divide both sides by 3 to completely isolate 'x':
\[\frac{3x}{3} > \frac{2}{3}\]
Which simplifies to:
\[x > \frac{2}{3}\]
By dividing by 3, we've isolated 'x', given us the solution. Notice that dividing by a positive number doesn't change the inequality direction.
However, be cautious: if you ever divide or multiply by a negative number, you must flip the inequality sign. Always check your work to ensure the logical flow of the solution.

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Most popular questions from this chapter

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