Problem 25 - Algebra and Trigonometry

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 4-3 x \leq-(1+8 x) $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -1]\).

Step 1: Distribute Negative Sign

First, let's simplify the right side of the inequality by distributing the negative sign. The inequality becomes: \[ 4 - 3x \leq -1 - 8x \].

Step 2: Move All Terms Involving x to One Side

Add \(8x\) to both sides of the inequality to try and isolate \(x\). This results in: \[ 4 + 5x \leq -1 \].

Step 3: Isolate the Variable x

Subtract 4 from both sides to further isolate the term involving \(x\): \[ 5x \leq -5 \].

Step 4: Solve for x

Divide both sides of the inequality by 5 to solve for \(x\): \[ x \leq -1 \].

Step 5: Express in Interval Notation

The solution \(x \leq -1\) in interval notation is: \[ (-\infty, -1] \].

Step 6: Graph the Solution Set

The graph of the solution set \((-\infty, -1]\) includes a filled circle at \(-1\) (indicating the endpoint is included) and an arrow extending to the left toward \(-\infty\).

Key Concepts

Solving Inequalities

Let's dive into solving linear inequalities, which are mathematical statements comparing two expressions with inequality symbols like \( \,\leq, \,<, \,\geq, \,> \.\). These steps are similar to solving equations, but always remember the special rule when multiplying or dividing by a negative number: the inequality direction flips. Here, the goal was to solve the inequality \( 4 - 3x \leq -(1 + 8x) \). We started by distributing the negative sign on the right. This means multiplying the negative sign through each term in the parentheses. After the distribution, we got \( 4 - 3x \leq -1 - 8x \).Next, consolidate all terms with \(x\) on one side, which often involves simple addition or subtraction. In our problem, we added \(8x\) to both sides, leading to \( 4 + 5x \leq -1 \). Finally, isolate \(x\) by algebraic manipulation, removing constants or coefficients till you solve for \(x\), giving us the solution \(x\leq -1\).

Interval Notation

Understanding interval notation is crucial for expressing the solutions of inequalities. It's a concise way to denote numbers lying between a range on the number line.For \(x \leq -1\), interval notation is \(( -\infty, -1 ]\). How does it work?

• The parentheses \(( \, \) or \()\) show that an endpoint is not included.
• The bracket \([ \) symbol indicates that an endpoint is included.
• \(-\infty\) always pairs with a parenthesis because infinity isn't a specific number we can "reach."

In our problem, \(-1\) is included (as it fulfills the inequality \(x \leq -1\)), so we use a bracket. Thus, our solution in interval notation is \((-\infty, -1]\).

Graphing Solutions

Graphing the solution set helps visualize the range of values that satisfy the inequality. For \(x \leq -1\), we represent solutions on a number line.To graph \((-\infty, -1]\):

• Draw a number line.
• Place a filled circle (or dot) on \(-1\). This signifies that \(-1\) is included in the solution set (confirmed by the bracket in interval notation).
• Draw an arrow extending leftward from \(-1\) towards \(-\infty\) to show all numbers less than \(-1\) are part of the solution.

This visual representation complements the algebraic solution, providing another layer of understanding by showing how \(x\) must be less than or equal to \(-1\).

Distributive Property

The distributive property is a fundamental concept in algebra, stating that \(a(b + c) = ab + ac\). It essentially allows you to "distribute" multiplication over addition or subtraction inside parentheses.In the inequality \(4 - 3x \leq -(1 + 8x)\), the distributive property helped simplify the right side. The expression \(-(1 + 8x)\) is distributed as:

• \(-1 \times 1 = -1\)
• \(-1 \times 8x = -8x\)

Resulting in: \(4 - 3x \leq -1 - 8x\).Using the distributive property effectively simplifies expressions, making it easier to isolate variables and solve inequalities.