Problem 171 Solve. Graph all solutions on a ... [FREE SOLUTION]

Chapter 1: Problem 171

Solve. Graph all solutions on a number line and provide the corresponding interval notation. $$ -4 \leq 3 x+5<11 $$

Short Answer

Expert verified

The solution is $[-3, 2)$ with $-3$ included and $2$ not included.

Step by step solution

Break Down the Compound Inequality

The compound inequality is given as \[ -4 \leq 3x + 5 < 11. \]This means we have two inequalities to solve: 1. $ -4 \leq 3x + 5 $2. $ 3x + 5 < 11 $.

Solve the First Inequality

Take the first part of the compound inequality:\[ -4 \leq 3x + 5. \]Subtract 5 from both sides:\[ -4 - 5 \leq 3x + 5 - 5 \quad \Rightarrow \quad -9 \leq 3x. \]Divide both sides by 3 to isolate $ x $:\[ \frac{-9}{3} \leq x \quad \Rightarrow \quad -3 \leq x. \]

Solve the Second Inequality

Now, solve the second part of the compound inequality:\[ 3x + 5 < 11. \]Subtract 5 from both sides:\[ 3x + 5 - 5 < 11 - 5 \quad \Rightarrow \quad 3x < 6. \]Divide both sides by 3 to find $ x $:\[ x < \frac{6}{3} \quad \Rightarrow \quad x < 2. \]

Combine the Solutions

We have two inequalities now:\[ -3 \leq x \quad \text{and} \quad x < 2. \]These can be combined to form the compound inequality:\[ -3 \leq x < 2. \]

Graph the Solutions

Graph the solution on a number line. Draw a closed circle at $-3$ to indicate that $-3$ is included in the solutions ($\leq$), and an open circle at $2$ to indicate that $2$ is not included ($<$). Shade the entire region between $-3$ and $2$ to represent all the values that satisfy the inequality.

Write the Interval Notation

The interval notation representing the solution $-3 \leq x < 2$ is written as:\[ [-3, 2). \]This denotes all numbers from $-3$ to $2$, including $-3$ but not $2$.

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Key Concepts

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

Number Line Graph

A number line graph provides a visual representation of inequalities and can help you better understand the range of solutions. In our original problem, we are given the compound inequality $-3 \leq x < 2$. To correctly graph this on a number line, follow these steps:

Start by drawing a horizontal line with evenly spaced intervals.
Mark the points $-3$ and $2$ on the line, as these are the boundaries of your solution.
Since $-3$ is included in the solution set (indicated by $\leq$), represent this with a closed circle on $-3$.
Since $2$ is not included in the solution set (indicated by $<$), represent this with an open circle on $2$.
Shade the region between the closed circle on $-3$ and the open circle on $2$ to demonstrate all the values of $x$ that satisfy the inequality.

Number line graphs are crucial for visualizing solutions and can quickly show you which values are included or excluded from the solution set.

Interval Notation

Interval notation offers a shorthand method for writing a range of numbers defined by inequalities, making it much simpler to understand and use. For the inequality $-3 \leq x < 2$, the interval notation is written as $[-3, 2)$. Here's why:

The square bracket $[$ on $-3$ indicates that $-3$ is included in the solution (known as a 'closed interval').
The parenthesis $)$ on $2$ means $2$ is not included in the solution (an 'open interval').
The comma separates the lower and upper bounds of the interval.

This notation is compact and very useful in mathematical writing. It provides an efficient way to express which values are part of the solution set without having to describe them in words every time.

Inequality Solutions

Solving inequalities involves finding all possible values of a variable that satisfy the given inequality conditions. For our problem, we started with the compound inequality $-4 \leq 3x+5 < 11$. This meant finding values of $x$ that satisfy both parts of this compound inequality:

The first step was to break the compound inequality into two separate inequalities: $-4 \leq 3x + 5$ and $3x + 5 < 11$.
Each inequality was solved separately, by first isolating $3x$, then dividing by $3$ to solve for $x$.
For the inequality $-4 \leq 3x + 5$, we ended up with $-3 \leq x$.
For $3x + 5 < 11$, we found $x < 2$.
By combining these, the solution to the original compound inequality was $-3 \leq x < 2$.

The process of solving inequalities often includes simplifying each inequality, combining solutions, and then representing them. This lays the foundation for interpreting more complex mathematical relationships in future topics.

91影视

Solve. Graph all solutions on a number line and provide the corresponding interval notation. $$ -4 \leq 3 x+5<11 $$

Short Answer

Step by step solution

Break Down the Compound Inequality

Solve the First Inequality

Solve the Second Inequality

Combine the Solutions

Graph the Solutions

Write the Interval Notation

Key Concepts

Number Line Graph

Interval Notation

Inequality Solutions

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