Problem 35 - Intermediate Algebra

Question

Solve each double inequality. Graph the solution set and write it using interval notation. \(0.9<2 x-0.7<1.5\)

Step-by-Step Solution

Answer

Interval notation: \((0.8, 1.1)\)

Step 1: Analyze the Double Inequality

First, identify the inequality to solve: \(0.9 < 2x - 0.7 < 1.5\). This is a double inequality, meaning there are two solutions that satisfy both inequalities simultaneously.

Step 2: Solve the Left Inequality

Consider the first part of the inequality: \(0.9 < 2x - 0.7\). Add \(0.7\) to both sides to isolate the term with \(x\). This gives us \(0.9 + 0.7 < 2x\), simplifying to \(1.6 < 2x\). Divide both sides by 2, resulting in \(0.8 < x\).

Step 3: Solve the Right Inequality

Now, take the second part of the inequality: \(2x - 0.7 < 1.5\). Add \(0.7\) to both sides to get \(2x < 1.5 + 0.7\), which simplifies to \(2x < 2.2\). Divide both sides by 2 to isolate \(x\), yielding \(x < 1.1\).

Step 4: Combine the Solutions

Combine the solutions from both inequalities: \(0.8 < x < 1.1\). This means that \(x\) is greater than \(0.8\) and less than \(1.1\).

Step 5: Graph the Solution

To graph the solution, draw a number line and mark open circles at \(0.8\) and \(1.1\) since these values are not included in the solution set. Shade the region between \(0.8\) and \(1.1\) to indicate all the numbers in this interval are solutions.

Step 6: Write in Interval Notation

Express the solution in interval notation. Since \(x\) is between \(0.8\) and \(1.1\), use parentheses to represent that \(0.8\) and \(1.1\) are not included: \((0.8, 1.1)\).

Key Concepts

Interval Notation

Interval notation is a compact way to express a range of values that satisfy an inequality. It's especially useful for double inequalities, where we often have a continuous range of solutions. In interval notation, parentheses and brackets have specific meanings:

• Parentheses, \( ... \), indicate that the endpoints are not included in the solution set. For example, \(0.8, 1.1\) represents all numbers between 0.8 and 1.1, excluding 0.8 and 1.1 themselves.
• Brackets, \[ ... \], mean that the endpoints are included. For example, \[0.8, 1.1\] means the range includes 0.8 and 1.1 as well as all numbers in between.

In the context of a double inequality like \(0.9 < 2x - 0.7 < 1.5\), once solved, we find the solution is \(0.8 < x < 1.1\). Since 0.8 and 1.1 are not included, we use parentheses in interval notation, resulting in \(0.8, 1.1\). This denotes that all numbers greater than 0.8 and less than 1.1 satisfy the inequality.

Graphing Inequalities

Graphing inequalities involves representing the solution of an inequality on a number line. This visual representation helps you understand the range of values that make an inequality true. When graphing:

• Draw a number line and identify important points, such as the boundaries of the solution range.
• Use open circles to indicate values that are not included in the solution. For example, for \(x > 0.8\), you'd place an open circle at 0.8.
• Shade the region of the number line where the solution lies. This shows which values satisfy the inequality.

In our solved problem, we graph \(0.8 < x < 1.1\) by marking open circles at 0.8 and 1.1, with shading between these points. This indicates all the numbers between 0.8 and 1.1 are solutions, and because the circles are open, neither 0.8 nor 1.1 themselves are included.

Solving Inequalities

Solving inequalities often requires similar steps as solving equations, with a few additional considerations to determine the allowable range of values. Here's the basic process:

• Isolate the variable term on one side. For double inequalities, solve separately for each part and ensure the inequality sign direction remains consistent.
• If you multiply or divide by a negative number, remember to reverse the inequality sign.
• Combine the solutions from each part to find the range of valid solutions.

For the inequality \(0.9 < 2x - 0.7 < 1.5\), we first solve \(0.9 < 2x - 0.7\) and \(2x - 0.7 < 1.5\) separately:

• Add 0.7 to both sides of \(0.9 < 2x - 0.7\) to get \(1.6 < 2x\). Dividing by 2, we find \(0.8 < x\).
• For \(2x - 0.7 < 1.5\), add 0.7 to yield \(2x < 2.2\), and dividing by 2 gives \(x < 1.1\).

These steps lead us to the combined solution \(0.8 < x < 1.1\), which is then expressed in interval notation as \(0.8, 1.1\). This procedure not only finds the solution but also verifies that both parts of the inequality are satisfied simultaneously.