Problem 4 Graph the inequality \(-3 \leq x... [FREE SOLUTION]

Chapter 7: Problem 4

Graph the inequality \(-3 \leq x-5<5\).

Short Answer

Expert verified

Answer: The complete solution of the compound inequality is \(2 \leq x < 10\). It can be represented on a number line by placing a closed circle at 2, an open circle at 10, and connecting the circles with a line to represent the values between 2 and 10 where the inequality holds true.

Step by step solution

Solve the first inequality

First, let's solve the inequality \(-3 \leq x-5\). Our goal is to isolate \(x\) on one side of the inequality. \begin{align*} -3 &\leq x-5 \\ -3 + 5 &\leq x \\ 2 &\leq x \end{align*} So, the solution for the first inequality is \(x \geq 2\).

Solve the second inequality

Now, let's solve the second inequality \(x-5<5\). Again, we want to isolate \(x\) on one side of the inequality. \begin{align*} x-5 &< 5 \\ x &< 5+5 \\ x &< 10 \end{align*} So, the solution for the second inequality is \(x < 10\).

Find the intersection of the solutions

We're looking for values of \(x\) that satisfy both inequalities, so we want the intersection of the two solutions. The first inequality has the solution \(x \geq 2\), and the second inequality has the solution \(x < 10\). Therefore, the solution of the compound inequality is \(2 \leq x < 10\).

Graph the solution on a number line

Now that we've found the complete solution, let's represent it on a number line. Draw a number line with numbers from 0 to 12, marking 2 and 10 in particular. Since the inequality includes the value 2, we will use a closed circle at 2. Since the solution does NOT include 10, we will use an open circle at 10. Finally, draw a line connecting the two circles, representing the values between 2 and 10 where the inequality holds true.

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Key Concepts

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

Compound Inequalities

Compound inequalities are algebraic expressions where two inequalities are combined into one statement by the words 'and' or 'or'. It's like a mathematical sandwich, where you have a lower bound and an upper bound, and we are interested in the numbers that satisfy both conditions at the same time.

When using 'and' to connect two inequalities, it means we're looking for numbers that satisfy both inequalities simultaneously. In the given exercise, we have the compound inequality \( -3 \leq x-5 < 5 \), which translates to finding all values of \(x\) that make both \( -3 \leq x-5 \) and \( x-5 < 5 \) true at the same time.

Number Line

A number line is a visual representation of numbers laid out in a straight line, where each point corresponds to a number. For graphing inequalities, a number line helps us see the set of solutions at a glance.

On a number line for the exercise \( -3 \leq x-5 < 5 \), we use a closed circle at \( x = 2 \) to indicate that 2 is included in the solution set (\( x \geq 2 \) indicates '2 or greater') and an open circle at \( x = 10 \) to show that 10 is not included in the possible values of \(x\) (\( x < 10 \) means 'less than 10'). The part of the line between your two markers represents all the numbers that fulfill the compound inequality.

Algebraic Solution

Finding the algebraic solution to inequalities involves isolating the variable (traditionally \(x\)) on one side of the inequality to determine its possible values. This process often includes simple arithmetic operations like addition, subtraction, multiplication, and division, while remembering to flip the inequality sign if we multiply or divide by a negative number.

In our example, solving the two parts of the compound inequality \( -3 \leq x-5 < 5 \) led us to the algebraic solution \(2 \leq x < 10\). Algebraic solutions provide the basis for understanding the range of values that variables within inequalities can take.

Inequality Notation

Inequality notation conveys the relationship between values using symbols such as \( < \) (less than), \( > \) (greater than), \( \leq \) (less than or equal to), and \( \geq \) (greater than or equal to). These symbols help to define the set of numbers that are solutions to an inequality.

In the context of the exercise, \( -3 \leq x-5 < 5 \) uses both \( \leq \) and \( < \) to inform us that \(x\) can be equal to or greater than 2 but must be strictly less than 10. Recognizing and understanding these symbols is crucial for solving and graphing inequalities effectively.

91影视

Graph the inequality \(-3 \leq x-5<5\).

Short Answer

Step by step solution

Solve the first inequality

Solve the second inequality

Find the intersection of the solutions

Graph the solution on a number line

Key Concepts

Compound Inequalities

Number Line

Algebraic Solution

Inequality Notation

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