Problem 39 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq 6\), presented in interval notation as \([6, \infty)\).

Step 1: Simplify the inequality

Begin by simplifying the left-hand side of the inequality which is \(1-(x+3)\). This can be rewritten as \(1-x-3\) or \(-x-2\). Now, the inequality becomes \(-x - 2 \geq 4 - 2x\).

Step 2: Solve for x

Next, start to isolate \(x\) by adding \(2x\) to both sides of the inequality to cancel out \(-2x\) on the right-hand side. Then the inequality becomes \(x - 2 \geq 4\).

Step 3: Continue to solve for x

Now, to completely isolate \(x\), add 2 to both sides of the inequality, cancelling out \(-2\) on the left-hand side. The inequality now is \(x \geq 6\).

Step 4: Write solution in interval notation

In interval notation, \(x \geq 6\) can be written as \([6, \infty)\), which means that \(x\) includes all numbers from 6 up to positive infinity.

Step 5: Graph the solution on the number line

On a number line, draw a solid circle at 6 because the value 6 is included in the solution set, and then draw an arrow extending to the right from 6 towards positive infinity, indicating all the numbers greater than or equal to 6 are in the solution set.