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Chapter 1: Problem 35

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$4(x+1)+2 \geq 3 x+6$$

Short Answer

Expert verified
The solution is \(x \geq 0\), expressed in the interval notation as \([0, \infty)\).

Step by step solution

01

Distributive Property

Start by using the distributive property, which will allow the equation to be simplified. The distributive law allows us to multiply 4 through the parentheses to get \(4x + 4 + 2 \geq 3x + 6\).
02

Simplification

Next, simplify the equation. Combine the like terms on the left side of the inequality to get \(4x + 6 \geq 3x + 6\).
03

Solving the Inequality

Subtract \(3x\) from both sides, and also subtract 6 from both sides. This results in \(x \geq 0\).
04

Interval Notation

Express the solution \(x \geq 0\) in interval notation, which is \([0, \infty)\). The square bracket indicates that the number is included in the interval, while the parentheses indicates that the end point (infinity) is not included.
05

Number Line Graph

Graph the solution on a number line. Draw a solid dot at 0, indicating that it is included in the solution set. Then, draw a line extending to the right from 0, towards positive infinity, indicating that all numbers greater than 0 are part of the solution set.

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