Problem 16 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 32 , find the solution set of the given inequality and illustrate the solution on the real number $$ 2 \leq 5-3 x<11 $$

Step-by-Step Solution

Answer

The solution set is \(-2 < x \leq 1\)

Step 1 - Break the Compound Inequality

The given inequality is a compound inequality: \[2 \leq 5-3x < 11\]This can be split into two separate inequalities:\[2 \leq 5 - 3x\] and \[5 - 3x < 11\]

Step 2 - Solve the First Inequality

Start with the first inequality \[2 \leq 5 - 3x\] Subtract 5 from both sides:\[2 - 5 \leq -3x\]Simplify:\[-3 \leq -3x\]Divide by -3 and remember to reverse the inequality sign:\[\frac{-3}{-3} \geq \frac{-3x}{-3}\]\[1 \geq x\] or \[x \leq 1\]

Step 3 - Solve the Second Inequality

Now, solve the second inequality:\[5 - 3x < 11\] Subtract 5 from both sides:\[5 - 3x - 5 < 11 - 5\]Simplify:\[-3x < 6\]Divide by -3 and remember to reverse the inequality sign:\[\frac{-3x}{-3} > \frac{6}{-3}\]\[x > -2\]

Step 4 - Combine the Results

Combine the results from both inequalities:\[x > -2\] and \[x \leq 1\]Therefore, the solution set is:\[-2 < x \leq 1\]

Step 5 - Illustration on the Number Line

Draw a number line and plot the interval \[-2 < x \leq 1\]:- Place an open circle at -2 (not included) and a closed circle at 1 (included).- Shade the region between -2 and 1.

Key Concepts

Inequality Solution Set

An inequality solution set is the range of values that satisfy an inequality. In our given exercise, we start with the compound inequality: \ \[2 \leq 5 - 3x < 11\] \
To find the solution set, we break it down into two separate inequalities: \ \[2 \leq 5 - 3x\] and \[5 - 3x < 11\]. \
Solving each inequality separately, we first address \[2 \leq 5 - 3x\]. By isolating x, we find that \[x \leq 1\]. Next, solving \[5 - 3x < 11\], we get \[x > -2\]. \
Combining both results, the solution set becomes \[-2 < x \leq 1\]. This represents all the values of x that satisfy both inequalities simultaneously.

Number Line Illustration

Visualizing the solution set on a number line is crucial for understanding inequalities. For the solution set \[-2 < x \leq 1\], we use the following steps: \
- Draw a number line and mark the critical points, -2 and 1. \
- Place an **open circle** at -2 to indicate -2 is not included (x is strictly greater than -2). \
- Place a **closed circle** at 1 to indicate 1 is included (x is less than or equal to 1). \
- Shade the region between -2 and 1 to show all the values that x can take. \
This clear visual aid helps one quickly grasp the range of solutions for the inequality.

Solving Inequalities

Solving inequalities involves a sequence of logical steps to isolate the variable and determine its range of values. Here’s a breakdown of solving our sample compound inequality: \
1. **Break Down the Inequality:** Separate the compound inequality into two inequalities. \
2. **Solve Individual Inequalities:** For each inequality, perform arithmetic operations to isolate the variable. Remember to flip the inequality sign when multiplying or dividing by a negative number. \[2 - 5 \leq -3x\] simplifies to \[-3 \leq -3x\], and dividing by -3 changes the sign, resulting in \[x \leq 1\]. Similarly, solving \[5 - 3x < 11\], you end up with \[x > -2\]. \
3. **Combine Results:** The final solution set is the intersection of the solutions from each inequality. This step typically involves using the logical AND condition. In our example, this results in \[-2 < x \leq 1\]. \
By following these steps, solving inequalities becomes a straightforward task. Each step ensures you are narrowing down the range of possible values for the variable.