Problem 29 - Algebra and Trigonometry

Question

\(9-32\) me solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ -2<8-2 x \leq-1 $$

Step-by-Step Solution

Answer

The solution is \([4.5, 5)\), graphically a solid dot at 4.5 and an open dot at 5, with shading in between.

Step 1: Isolate the Variable Part

First, start by isolating the term with the variable in the inequality. Begin by subtracting 8 from all parts of the inequality: \[-2 - 8 < 8 - 2x - 8 \leq -1 - 8\]. Simplify this to get: \[-10 < -2x \leq -9\].

Step 2: Solve for x

Next, we need to solve for \(x\). To do this, divide each part of the inequality by \(-2\). Remember, when dividing or multiplying inequalities by a negative number, the inequality signs must be reversed:\[\frac{-10}{-2} > \frac{-2x}{-2} \geq \frac{-9}{-2}\]. This simplifies to \[5 > x \geq 4.5\].So, \(x\) is in the interval \([4.5, 5)\).

Step 3: Express the Solution in Interval Notation

Rewrite the solution in interval notation. Since \(x\) is greater than or equal to 4.5 and less than 5, we use a closed bracket on 4.5 and an open bracket on 5: \([4.5, 5)\).

Step 4: Graph the Solution Set

To graph the solution set on a number line, draw a line and mark the points 4.5 and 5. Include a solid dot at 4.5 to indicate that it is part of the solution set, and an open dot at 5 to indicate that it is not included. Shade the region between these two points to represent all values that \(x\) can take.

Key Concepts

solving inequalities

Solving inequalities is a foundational skill in mathematics, much like solving equations, but with a few extra rules. In an inequality, instead of having an equal sign, you might have symbols like ">", "<", "≥", or "≤". These symbols show the range of values the variable can take. Let's break down the process using our problem for clarity.

Our task is to solve \(-2 < 8 - 2x \leq -1\). It means finding the values of \(x\) that make this statement true. We start solving by isolating terms with variables. Usually, this involves moving all terms containing the variable to one side of the inequality.

• Subtract 8 from each part: \(-2 - 8 < 8 - 2x - 8 \leq -1 - 8\), simplifying to \(-10 < -2x \leq -9\).
• To isolate \(x\), divide everything by \(-2\). Remember, dividing or multiplying an inequality by a negative reverses the inequality sign. This gives \(5 > x \geq 4.5\).

Thus, we find that \(x\) is between 4.5 and 5, but be careful with the direction of the inequality arrows.

interval notation

Once we solve an inequality, the next step is to express its solution in interval notation. This notation is a simple way of writing solutions to inequalities. It indicates the set of all possible values for \(x\).

• For our solution, \(5 > x \geq 4.5\), we need to write this in interval notation.
• The square bracket "[" means that the endpoint is included, while the round bracket ")" means it is not included.

So, since \(x\) is greater than or equal to 4.5 (included) and less than 5 (not included), the interval notation is \([4.5, 5)\). This compact form is both precise and concise, which makes it handy for communicating solutions.

graphing inequalities

Graphing inequalities is a practical way to visualize solutions on a number line. It helps to see all the possible values a variable can take.

For the inequality \(5 > x \geq 4.5\), we represent this graphically by marking and shading parts of a number line:

• First, identify the important points, which are 4.5 and 5.
• Place a solid dot on 4.5, showing it is part of the solution set, since 4.5 is included (denoted by the square bracket in interval notation).
• Place an open dot at 5 to show it is not included in the solution set (denoted by the round bracket).
• Shade the region between 4.5 and 5 to represent all values \(x\) can take.

This visualization allows you to easily see the solution set, making inequalities easier to understand and communicate.