Problem 62 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ 1<8 x+5<5 $$

Step-by-Step Solution

Answer

\(-\frac{1}{2} < x < 0\) or in interval notation: \((-\frac{1}{2}, 0)\).

Step 1: Identify the Inequalities

The given double inequality can be split into two separate inequalities: \(1 < 8x + 5\) and \(8x + 5 < 5\). We need to solve each inequality individually.

Step 2: Solve the First Inequality

To solve \(1 < 8x + 5\), we first subtract 5 from both sides to get \(1 - 5 < 8x\). This simplifies to \(-4 < 8x\). Next, divide both sides by 8 to find \(x\). Thus, \(-\frac{1}{2} < x\).

Step 3: Solve the Second Inequality

For the inequality \(8x + 5 < 5\), subtract 5 from both sides to get \(8x < 0\). Then, divide both sides by 8 to solve for \(x\). This results in \(x < 0\).

Step 4: Combine the Solutions

To find the solution set for the combined inequalities \(-\frac{1}{2} < x < 0\), we take the intersection of the solutions from Step 2 and Step 3. This results in \(-\frac{1}{2} < x < 0\).

Step 5: Represent Solution in Interval Notation

The interval notation for the solution \(-\frac{1}{2} < x < 0\) is \((-\frac{1}{2}, 0)\).

Step 6: Graph the Solution Set on a Number Line

Draw a number line, marking \(-\frac{1}{2}\) and \(0\) with open circles (since they are not included in the solution set). Shade the region between \(-\frac{1}{2}\) and \(0\) to represent the solution \((-\frac{1}{2}, 0)\).

Key Concepts

Inequalities

Inequalities are mathematical expressions that describe relationships of inequality rather than equality between two values. The solution to an inequality is the set of all values that make the inequality true. In our exercise, we are dealing with a compound inequality, specifically a double inequality: \(1 < 8x + 5 < 5\). To solve this, we manage the expression as two separate inequalities:

• \(1 < 8x + 5\)
• \(8x + 5 < 5\)

This involves finding the set of values of \(x\) that satisfies both conditions simultaneously, allowing us to determine a range of solutions rather than a single answer. This makes inequalities different from equations, where typically one exact solution or a finite set of solutions exists.

Solution Set

The solution set of an inequality is essentially the range of values that satisfy the inequality. For double inequalities like \(-\frac{1}{2} < x < 0\), it represents all values \(x\) that lie within these limits.

Solving individual inequalities, \(1 < 8x + 5\) and \(8x + 5 < 5\), leads us to derive the two conditions:

• \(-\frac{1}{2} < x\)
• \(x < 0\)

Combining these gives the solution set \(-\frac{1}{2} < x < 0\). When describing this range in interval notation, where endpoints are given special attention, we express it as \(( -\frac{1}{2}, 0 )\). Here, both endpoints are excluded from the solution set, hence the use of parentheses rather than brackets.

Graphing on Number Line

Graphing the solution set of an inequality on a number line provides a visual representation of the solution. For our inequality \(-\frac{1}{2} < x < 0\), this involves:

• Drawing a horizontal line with tick marks at \(-\frac{1}{2}\) and \(0\)
• Using open circles at \(-\frac{1}{2}\) and \(0\) to indicate that these points are not part of the solution set
• Shading the line segment between \(-\frac{1}{2}\) and \(0\) to show where \(x\) values lie

This graphical method helps reinforce the concept that the solution is a continuous range of values rather than discrete points. It's important to note this distinction, as open circles on a number line mean the respective points are not included in the solution, while closed circles would imply inclusion.