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Chapter 2: Problem 45

Solve each inequality and graph the solution on the number line. $$0 \leq x+10 \leq 10$$

Short Answer

Expert verified
The solution to the compound inequality \(0 \leq x+10 \leq 10\) is \(-10 \leq x \leq 0\). When graphed on a number line, this is represented as a closed circle at -10 and 0, with all the numbers in between shaded.

Step by step solution

01

Solve the Left Inequality

The compound inequality is \(0 \leq x+10 \leq 10\). Start by isolating \(x\) in the left inequality: \(0 \leq x+10\). Subtract 10 from both sides: \(x \geq -10\).
02

Solve the Right Inequality

Next, isolate \(x\) in the right inequality: \(x+10 \leq 10\). Subtract 10 from both sides: \(x \leq 0\).
03

Combine the Results

Combine the results from the left and right inequalities. Since \(x\) has to satisfy both inequalities at the same time, take the intersection of both inequalities, which results in \(-10 \leq x \leq 0\).
04

Graph the Solution on the Number Line

The solution describes numbers that are between -10 and 0, inclusive. To plot this on a number line, make a closed circle at -10 and 0, and shade everything in between. The closed circle indicates that -10 and 0 are included in the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Compound Inequalities
Compound inequalities involve finding the values of a variable that satisfy more than one inequality at the same time. These can be thought of as two separate inequalities that are combined into one statement.
  • In a compound inequality such as \(0 \leq x+10 \leq 10\), both conditions must be true at the same time.
  • We split this compound inequality into two parts: \(0 \leq x+10\) and \(x+10 \leq 10\).
Solving such inequalities involves breaking them down. First, deal with each inequality on its own. Then, find a common solution that satisfies both at once. This approach ensures that the solution to the compound inequality respects all given constraints.
Visualizing Solutions with Number Line Graphing
Graphing inequalities on a number line helps visualize where the possible solutions fall. A number line is a straightforward tool to express real number solutions.
  • To graph from the inequality \(-10 \leq x \leq 0\), locate these two endpoints on the line.
  • Place a closed circle over -10 and 0. Closed circles denote that these endpoints are part of the solution set.
  • Shade the portion of the line between -10 and 0. This shaded area represents all the numbers \(x\) can take, including -10 and 0.
Graphing the solution gives you an immediate and clear idea of the range of values that \(x\) can have. This visual tool makes the concept of inequalities more tangible and easier to interpret.
Exploring the Intersection of Inequalities
The intersection of inequalities refers to the common values that satisfy all parts of a compound inequality. This arises when we combine results from separate inequalities into a single solution.
  • In our example, we had two inequalities: \(x \geq -10\) and \(x \leq 0\).
  • The intersection is where these two solution sets overlap.

To find this overlap, think of it as looking for numbers shared by both solutions.
  • Here, the overlap is all numbers from -10 to 0.
  • Thus, the compound solution becomes \(-10 \leq x \leq 0\).
Understanding this intersection is crucial because it is the essence of solving compound inequalities. It represents the common ground where all conditions meet. This ensures you have a valid and complete solution.

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