Problem 39 - College Algebra

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(1-(x+3) \geq 4-2 x\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq 6\), which is expressed in interval notation as \([6,\infty)\). On a number line, this is represented by a filled circle at 6 and a line extending to the right towards positive infinity.

Step 1: Simplify the Given Inequality

We begin by simplifying \(1-(x+3) \geq 4-2x\). Distributing the negative sign in \(1-(x+3)\) yields \(1 - x - 3 \geq 4 - 2x\), which on further simplification becomes \(-x - 2 \geq 4 - 2x\).

Step 2: Solve the Inequality

Next we solve the inequality. Isolate \(x\) by adding \(2x\) to both sides to cancel the \(x\) term on the right side, which gives us \(-x - 2 + 2x \geq 4\). Simplifying gives \(x - 2 \geq 4\). We then add \(2\) to both sides which results in \(x \geq 6\).

Step 3: Express the Solution in Interval Notation

The solution \(x \geq 6\) in interval notation becomes \([6,\infty)\). It means that \(x\) can be any number from 6, including 6, to positive infinity.

Step 4: Graph the Solution Set on a Number Line

To represent this solution set on a number line, a filled circle should be drawn at 6 (since it includes 6) and then a line drawn to the right extending towards positive infinity.

Key Concepts

Interval Notation

Interval notation is a mathematical method for representing a range of numbers. It’s a shorthand way to describe a set of real numbers between two endpoints. When we are solving linear inequalities, the solutions are often expressed as an interval to show all the numbers that satisfy the inequality.

For example, if we have the inequality solution as \( x \geq 6 \), using interval notation we write this as \( [6, \infty) \). The square bracket, `[`, tells us that 6 is included in the solution set (it's a closed interval), while the parenthesis, `)`, indicates that infinity is not an actual number we can reach but rather a concept that means the numbers go on indefinitely (it's an open interval). Understanding interval notation is crucial as it provides a clear and concise way to communicate the solution of inequalities.

Number Line Representation

Number line representation is a visual way to display the solution set of an inequality on a straight line that represents all real numbers. Each point on the line corresponds to a number, where points to the right represent greater numbers, and points to the left represent lesser numbers.

To graph the inequality \( x \geq 6 \), we start by putting a filled circle on the number 6 to indicate that it’s included in our solution set. Then, we draw a line extending to the right towards infinity to represent all numbers greater than or equal to 6. This visual aid helps students quickly see the range of numbers that satisfy the inequality and is especially useful for understanding concepts related to inequality solutions.

Inequality Simplification

Inequality simplification involves reducing an inequality to its simplest form to make it easier to solve. This process typically includes distributing multiplication across parentheses, combining like terms, and moving terms to one side of the inequality to isolate the variable.

In our exercise, we start by distributing the negative sign across \((x+3)\), which changes the problem to \(1 - x - 3 \geq 4 - 2x\). Then we combine like terms, resulting in \(-x - 2 \geq 4 - 2x\). By simplifying inequalities step by step, we effectively make the solution process more transparent and manageable, enabling students to reach the solution with greater ease.

Algebraic Solution Methods

Algebraic solution methods refer to the techniques used to find the value(s) that satisfy an equation or inequality. When working with inequalities, we aim to isolate the variable. This often involves performing the same operation on both sides of the inequality without changing its direction.

For the given inequality, we isolate the variable \(x\) by adding \(2x\) to both sides, which gives us \(x - 2 \geq 4\). Once isolated, we further simplify by adding 2 to both sides yielding \(x \geq 6\). Remember that if we multiply or divide both sides of an inequality by a negative number, we must reverse the inequality sign. It is these logical and systematic algebraic solution methods that provide a clear path to solving inequalities and understanding the mathematical reasoning behind them.