Problem 23 - Algebra and Trigonometry

Question

\(9-32\) me 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

Solution: \((-\infty, -1]\) with a solid dot on -1 on the number line.

Step 1: Distribute Negative Sign

First, simplify the right side of the inequality by distributing the negative sign across the expression inside the parentheses. This yields:\[-(1 + 8x) = -1 - 8x.\]

Step 2: Rewrite the Inequality

Substitute the simplified expression from Step 1 into the inequality. This gives:\[4 - 3x \leq -1 - 8x.\]

Step 3: Move All Variables to One Side

Add \(8x\) to both sides to move all \(x\) terms to the left:\[4 - 3x + 8x \leq -1.\]Simplify to:\[4 + 5x \leq -1.\]

Step 4: Isolate the Variable Term

Subtract 4 from both sides to isolate the \(x\) term:\[5x \leq -1 - 4.\]Thus:\[5x \leq -5.\]

Step 5: Solve for the Variable

Divide both sides by 5 to solve for \(x\):\[x \leq \frac{-5}{5}.\]This simplifies to:\[x \leq -1.\]

Step 6: Express the Solution in Interval Notation

The solution in interval notation is: \[(-\infty, -1].\] This represents all numbers less than or equal to -1.

Step 7: Graph the Solution Set

To graph the solution, draw a number line and shade all parts left of -1, including -1 itself. Use a solid dot at -1 to show that -1 is included in the solution.

Key Concepts

Interval Notation

Interval notation is a method used to express a range of numbers that are included in a solution set. It provides a concise way of writing a set of numbers without listing them out. In interval notation:

• Parentheses ")" or "(" are used to denote that an endpoint is not included in the interval (open interval).
• Brackets "]" or "[" are used to denote that an endpoint is included (closed interval).

For example, the solution \(x \leq -1\) is expressed in interval notation as \((-\infty, -1]\). This notation means that the solution includes all numbers less than or equal to -1. Here, \( -1 \) is included because a square bracket is used, whereas negative infinity always has a parenthesis because infinity is not a number that can be "reached."
This concise representation is very useful in mathematics, especially for linear inequalities, as it allows for a clear and precise definition of the range of solutions.

Solving Inequalities

Solving inequalities is the process of finding the values of a variable that satisfy an inequality. An inequality shows that two expressions are not equal, using symbols such as ">", "<", "≥", or "≤". Unlike equations, inequalities do not have just one solution; they often have a range of solutions.

Here are basic steps to solve linear inequalities:

• Simplify both sides of the inequality, if necessary. This often involves distributing and combining like terms.
• Use addition or subtraction to move variable terms to one side and constant terms to the other. This helps to isolate the variable term on one side of the inequality.
• Use multiplication or division to further simplify and solve for the variable. When multiplying or dividing by a negative number, remember to flip the inequality sign.

For example, solving \(4 - 3x \leq -(1 + 8x)\) involves several steps, such as distributing the negative, combining terms, and ultimately isolating \(x\). The final inequality \(x \leq -1\) provides the solution for the variable.

Graphing Solutions

Graphing the solutions of a linear inequality on a number line helps visualize the set of numbers satisfying the inequality. The graph provides a clear view of where the variable's solutions lie.To graph the inequality \(x \leq -1\):

• Draw a horizontal number line, and mark the point corresponding to \(-1\).
• Place a solid dot at \(-1\) to show that it is part of the solution set, since the inequality is "less than or equal to."
• Shade the region to the left of \(-1\). This indicates all numbers less than \(-1\), which are also included in the solution set.

The solid dot signifies that the number it's on is included in the solution, unlike an open dot which indicates the number is not included. This visual aid is especially beneficial for understanding the range of possible solutions in the context of inequalities.