Problem 38 - Algebra and Trigonometry

Question

In all exercises other than \(\varnothing\), use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ -4(x+2)>3 x+20 $$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < -4\). The solution set in interval notation is \(-\infty, -4)\).

Step 1: Simplify the inequality

First, distribute the -4 to both terms within the parentheses on the left side of the inequality, resulting to \(-4x-8>3x+20\).

Step 2: Isolate the variable

To isolate 'x', the equality should be manipulated. Subtract 3x and 8 from both sides of the inequality: \(-4x - 3x > 20 + 8\). This simplifies the inequality to \(-7x > 28\).

Step 3: Solve for x

Now, divide both sides of the inequality by -7 to solve for 'x'. When you divide or multiply an inequality by a negative number, you should flip the inequality sign. Hence, \(x < -4\).

Step 4: Represent the solution in interval notation and graph

The solution to the inequality in interval notation is \(-\infty, -4)\). The interval notation represents all the values 'x' can take on. In terms of a graph on a number line, a circle at -4 (not filled in because -4 is not included in the solution set), with an arrow pointing to the left represents all numbers less than -4.

Key Concepts

Interval Notation

Interval notation is a way to write the set of solutions for inequalities. It uses brackets and parentheses to describe intervals of numbers. In our exercise, we determined that the solution is all numbers less than \(-4\). This is written as \((-\infty, -4)\).

• Parentheses (like in \((-\infty, -4)\)) mean the endpoint is not included.
• Brackets would mean the endpoint is included (e.g., \([-4, \infty)\)).

Notice that negative infinity always gets a parenthesis because it's a concept, not a real number.

Solution Sets

A solution set includes all the possible values that make an inequality true. For the inequality \(-4(x+2)>3x+20\), we simplified and found that \(x < -4\).

• This means any number less than \(-4\) will satisfy the inequality.
• We express this entire range of solutions using interval notation as \((-\infty, -4)\).

This set captures every number in this range, making it a powerful way to understand all solutions at once.

Number Line Graphing

Graphing the solution on a number line helps visualize the set of solutions. For \(x < -4\), draw a number line and place an open circle on \(-4\) to indicate it's not included.

• An open circle means the specific number is not part of the solution.
• Draw an arrow pointing left from the circle to show all numbers less than \(-4\) are included.

This visual representation complements interval notation perfectly and makes understanding the solution much easier.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to isolate the variable. For the inequality \(-4(x+2)>3x+20\):

• First, distribute: \(-4x - 8 > 3x + 20\).
• Next, combine like terms by moving \(3x\) and \(-8\) across the inequality: \(-7x > 28\).
• Finally, divide by \(-7\): remember to flip the inequality to get \(x < -4\).

This process shows how critical each step is in solving inequalities. Ensuring the inequality sign flips when dividing by a negative preserves the truth of the inequality.