Problem 38 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$-4(x+2)>3 x+20$$

Step-by-Step Solution

Answer

The solution of the inequality \(-4(x+2) > 3 x+20\) is \(x < -4\). In interval notation, it is represented as \((-\infty, -4)\).

Step 1: Distribute on Both Sides

Distribute \(-4\) to both \(x\) and \(2\) on the left side of the inequality: \(-4x - 8 > 3x + 20\). This results in a simplified inequality.

Step 2: Move Terms to Isolate Variable

Move the \(x\) terms to one side and the numbers to the opposite side: \(-7x > 28\). This is accomplished by adding \(4x\) to both sides and subtracting \(20\) from both sides.

Step 3: Solve for x

Now we can solve for \(x\) by dividing every term by \(-7\). But remember, when you multiply or divide an inequality by a negative number, the greater than symbol will change to less than: hence \(x < -4\).

Step 4: Write in Interval Notation and Graph

Finally, represent the solution using interval notation: \((-\infty, -4)\), and then graph it on a number line. Since the less than symbol does not include the number \(-4\), use an open circle at \(-4\) on the graph.