Problem 35 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(4x + 4 + 2 \geq 3x + 6\).

Step 2: Simplification

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

Step 3: Solving the Inequality

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

Step 4: 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.

Step 5: 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.