Problem 22 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -8 x+1 \leq 29 $$

Step-by-Step Solution

Answer

The solution is \( x \geq -3.5 \), or in interval notation \([-3.5, \infty)\).

Step 1: Isolate the Variable Term

Start with the inequality \[ -8x + 1 \leq 29 \]Subtract 1 from both sides to isolate the term with the variable:\[ -8x + 1 - 1 \leq 29 - 1 \]This simplifies to:\[ -8x \leq 28 \]

Step 2: Solve for the Variable

To solve for \( x \), divide both sides of the inequality by \( -8 \). Remember, when you divide an inequality by a negative number, you must reverse the inequality sign:\[ x \geq -\frac{28}{8} \]Simplify the fraction:\[ x \geq -\frac{7}{2} \] or \( x \geq -3.5 \).

Step 3: Express the Solution Set in Interval Notation

The solution \( x \geq -3.5 \) means that \( x \) is any number greater than or equal to \(-3.5\). In interval notation, this is expressed as:\[ [-3.5, \infty) \]

Step 4: Graph the Solution Set

To graph the solution \( x \geq -3.5 \), draw a number line. Place a solid circle or closed bracket at \(-3.5\) to indicate that \(-3.5\) is included in the solution set. Draw an arrow going to the right, indicating all numbers greater than \(-3.5\) are part of the solution.

Key Concepts

Interval Notation

Interval notation is a concise way of expressing a range of values. It is especially useful in inequalities because it clearly shows the boundary values and whether they are included or excluded. In this exercise, after solving the inequality, we concluded that \( x \geq -3.5 \).

Here's how interval notation works for this example:

• **Square brackets** \([-3.5, \infty)\) indicate that the number \(-3.5\) is included in the solution set.
• **Parentheses** \([-3.5, \infty)\) signify that \(\infty\) is not a real number and thus can't be included, hence the parenthesis.

Interval notation thus conveys information compactly and efficiently. This entire set represents all real numbers from \(-3.5\) to infinity, with \(-3.5\) included.

Graphing Inequalities

Graphing inequalities involves showing our solution on a number line, which provides a visual representation of the set of solutions. For our inequality \(x \geq -3.5\), graphing becomes a straightforward task.

Here's how to graph this inequality:

• Draw a horizontal line, which represents all possible values for \(x\).
• Locate the number \(-3.5\) on this line.
• Because the inequality is \(\geq\), we will use a **solid circle** at \(-3.5\) to show that \(-3.5\) is included.
• Draw an arrow extending to the right from \(-3.5\) to illustrate that all numbers greater than \(-3.5\) are included.

This number line graph helps us visually confirm the solution set, identifying quickly which numbers satisfy the original inequality.

Variable Isolation

Variable isolation is the process of transforming an equation or inequality to solve for a specific variable. This often involves performing operations that "isolate" the variable on one side of the equation or inequality.

In our case, we started with the inequality \(-8x + 1 \leq 29\). Our goal is to "release" the \(x\) from other operations:

• First, subtract 1 from both sides to remove the constant term, leaving \(-8x \leq 28\).
• Second, divide by \(-8\) to solve for \(x\). Remember, dividing or multiplying by a negative flips the inequality sign, turning it into \(x \geq -3.5\).

This process is crucial, as isolating the variable allows us to express the inequality in a form where we can directly interpret the solution, leading to interval notation, and graphing as further steps.