Problem 40 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$5(3-x) \leq 3 x-1$$

Step-by-Step Solution

Answer

The solution to the linear inequality \(5(3-x) \leq 3x-1\) is \(x \geq 2\), which corresponds to the interval \([2, \infty)\) when expressed in interval notation. On the number line, this solution set is represented by a blackened circle on 2 and a line extending indefinitely to the right.

Step 1: Distribute the 5

Distribute the 5 to both terms within the parentheses: \(5*(3-x) = 15 -5x\)

Step 2: Rearrange the inequality

Organize the inequality so that like terms are on one side: \(15-5x \leq 3x - 1\). Then bring the x terms together on one side and the numerical terms on the other: \(15 +1 \leq 3x + 5x\). That simplifies to: \(16 \leq 8x\)

Step 3: Solve for x

By dividing each side by 8, we isolate x: \(x \geq 2\)

Step 4: Express the solution in interval notation

The solution is all x values greater than or equal to 2, expressed using interval notation: \([2, \infty)\)

Step 5: Graph the solution set on a number line

Mark a blackened circle on 2 (to represent inclusive of 2) and draw a line extending to the right (to represent all numbers greater than 2)