Problem 42 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -2(x-2)+14 x<7(2 x+1) $$

Step-by-Step Solution

Answer

The solution set is \(x > -\frac{3}{2}\), or \((-\frac{3}{2}, \infty)\).

Step 1: Distribute Terms on Both Sides

First, apply the distributive property to expand both sides of the inequality. For the left side, distribute \(-2\) over \((x-2)\) to get \(-2x + 4\). For the right side, distribute \(7\) over \((2x + 1)\) to get \(14x + 7\). The inequality becomes:\[-2x + 4 + 14x < 14x + 7\]

Step 2: Combine Like Terms

Next, combine like terms on the left side. Combine \(-2x\) and \(+14x\) to get \(12x\). This simplifies the inequality to:\[12x + 4 < 14x + 7\]

Step 3: Isolate Variable Terms

Move the \(14x\) term from the right to the left side by subtracting \(14x\) from both sides:\[12x - 14x + 4 < 7\] This simplifies to:\[-2x + 4 < 7\]

Step 4: Solve for x

Subtract \(4\) from both sides to isolate the term with \(x\):\[-2x < 3\]Now, divide both sides by \(-2\), remembering to flip the inequality sign:\[x > -\frac{3}{2}\]

Step 5: Express in Interval Notation

The solution set for the inequality \(x > -\frac{3}{2}\) is expressed in interval notation as:\((-\frac{3}{2}, \infty)\)

Step 6: Graph the Solution on a Number Line

Draw a number line. Use an open circle at \(-\frac{3}{2}\) to indicate that it is not included. Shade the line to the right to illustrate that all values greater than \(-\frac{3}{2}\) are solutions.

Key Concepts

Distributive Property

The distributive property is a crucial concept in solving inequalities and equations, as it allows us to simplify expressions by removing parentheses. When you distribute, you multiply the term outside the parentheses by each term inside. This process is essential to understanding how to move from complex, parenthesis-filled expressions to simpler, more manageable forms of an inequality.

For example, if you have \(-2(x-2)\), you apply the distributive property by multiplying \(-2\) with both \(x\) and \(-2\), resulting in \(-2x + 4\). Similarly, for the expression \(7(2x + 1)\), multiply \(7\) by both \(2x\) and \(1\) to get \(14x + 7\).

Once you distribute, it simplifies the problem because you can clearly see the coefficients of terms that can be combined, transferred, or rearranged to solve the inequality.

Combining Like Terms

Combining like terms is a vital process to streamline expressions and solve inequalities more efficiently. Think of it as gathering similar items together to keep things organized and simple.

When terms have the same variable to the same power, such as \(-2x\) and \(+14x\), they can be combined. In this case, \(-2x + 14x\) simplifies to \(12x\), reducing the inequality to a simpler form.

• Helps to simplify expressions.
• Makes equations easier to solve.
• Enables clearer visualization of the problem.

By combining like terms, you consolidate all the similar variables, which makes the subsequent steps of isolating variables and solving the inequality more straightforward.

Interval Notation

Interval notation is a concise way of representing a set of values that satisfy an inequality. It gives a clear range of solutions without having to graph them.

For the inequality \(x > -\frac{3}{2}\), the interval notation reads \((-\frac{3}{2}, \infty)\). This tells us that solutions for \(x\) include all real numbers greater than \(-\frac{3}{2}\), but not \(-\frac{3}{2}\) itself.

• A parenthesis \( ( \) or \( ) \) indicates that the endpoint is not included.
• Brackets would be used if an endpoint were included.

Example:

• \([-3, 5)\): includes \(-3\) but not \(5\).
• \((4, \infty)\): all numbers greater than \(4\).

Interval notation simplifies communication about solution sets and complements graphing solutions.

Number Line Graphing

Graphing the solution of an inequality on a number line is a visual representation that offers an intuitive understanding of the solution set.

To graph the solution \(x > -\frac{3}{2}\), start by drawing a number line. Follow these simple steps:

• Identify and mark the point \(-\frac{3}{2}\) on the line.
• Use an open circle at \(-\frac{3}{2}\) to indicate that it is not included in the solutions.
• Shade the region to the right of \(-\frac{3}{2}\), indicating that all values greater than \(-\frac{3}{2}\) satisfy the inequality.

This method provides an easy, straightforward view of which numbers are included in a solution set, making it a powerful tool for understanding inequalities.