Problem 35 - Algebra and Trigonometry

Question

In all exercises other than \(\varnothing\), use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ 4(x+1)+2=3 x+6 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0\).

Step 1: Distribute terms

Distribute 4 across \(x + 1\) in \(4(x+1)\) to get \(4x + 4\). The equation now reads \(4x + 4 + 2 = 3x + 6\).

Step 2: Combine like terms

Combine like terms on the left side of the equation to get \(4x + 6 = 3x + 6\).

Step 3: Solve for x

Subtract \(3x\) from both sides to get \(x + 6 = 6\). Then subtract 6 from both sides to get \(x = 0\).

Key Concepts

Interval Notation

Interval notation is a method used to describe the set of numbers that satisfy an inequality. It's a concise way to express the range of solutions without listing each individual number. This system is particularly useful because it clearly denotes the start and end of an interval, as well as whether these endpoints are included in the set.
For example, if an inequality solution includes all numbers greater than 3 but less than 5, it would be written as \( (3, 5) \). The parentheses indicate that 3 and 5 are not part of the solution.

Number Line Graphing

Graphing on a number line is a visual way to represent the solution to an inequality. A number line allows you to mark specific points and show ranges of values with ease. To graph an inequality, like \( x > -2 \) for example, you would draw an open circle at -2 on the number line to signify that -2 is not included in the solution set and then shade the line to the right of -2 to indicate all the numbers greater than -2 are included.

Combining Like Terms

Combining like terms is a key step in solving equations and inequalities. Like terms have the same variable raised to the same power. The process involves adding or subtracting coefficients (the numerical part of the terms) while keeping the variable part unchanged.
For instance, in the given inequality \(4x + 6 = 3x + 6\), 'like terms' are those with the variable \(x\) on both sides of the equation. By subtracting \(3x\) from \(4x\), you effectively combine these like terms to isolate \(x\) and solve for its value.