Chapter 1: Problem 62
Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. \(2-3 x \leq 5\)
Short Answer
Expert verified
\([-1, \infty)\)
Step by step solution
01
- Isolate the variable term
Start by moving the constant term on the left side to the right side of the inequality. Subtract 2 from both sides: \[2 - 3x - 2 \leq 5 - 2\]Simplifying, we get:\[-3x \leq 3\]
02
- Solve for the variable
To isolate \(x\), divide both sides of the inequality by -3. Remember to flip the inequality sign when dividing by a negative number: \[\frac{-3x}{-3} \geq \frac{3}{-3}\]This simplifies to:\[x \geq -1\]
03
- Express the solution in interval notation
The solution to \(x \geq -1\) can be written in interval notation as \[[-1, \infty)\].
04
- Graph the solution set
On a number line, draw a closed circle at -1 to indicate that -1 is included in the solution set. Then draw a line extending to the right to show that the solution includes all numbers greater than -1.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation is a way of expressing a range of values, commonly used to show the solution sets of inequalities. It helps to compactly represent all numbers between two endpoints, including whether the endpoints themselves are part of the solution.
For example, in the given problem, after solving the inequality, we found that the solution is all numbers greater than or equal to -1. In interval notation, this is written as \([-1, \infty)\).
The square bracket \[ \] means that -1 is included in the solution, while the parenthesis \( \) indicates that \(\infty\) is not a specific number but extends infinitely. Hence, all numbers starting from -1 up to infinity are part of the solution set using interval notation.
For example, in the given problem, after solving the inequality, we found that the solution is all numbers greater than or equal to -1. In interval notation, this is written as \([-1, \infty)\).
The square bracket \[ \] means that -1 is included in the solution, while the parenthesis \( \) indicates that \(\infty\) is not a specific number but extends infinitely. Hence, all numbers starting from -1 up to infinity are part of the solution set using interval notation.
Inequality Graphing
Graphing the solution of an inequality provides a visual representation of the solution set. It helps to see which numbers satisfy the inequality.
Here’s how to do it step-by-step:
Here’s how to do it step-by-step:
- First, identify and mark the key point of the solution. For our inequality \(x \geq -1\), this point is -1. You mark -1 on the number line.
- Next, choose the type of circle to use. Since -1 is included in the solution set (as indicated by the \geq sign), you place a closed circle at -1.
- Then, draw an arrow extending to the right from -1 to show that all numbers greater than -1 are included. This visualizes that the solution set extends infinitely in the positive direction.
Algebraic Manipulation
Algebraic manipulation is the process of restructuring expressions and equations to isolate variables and solve problems. It involves using fundamental algebraic operations such as addition, subtraction, multiplication, and division to rearrange terms.
In our problem, we had the inequality \(2 - 3x \leq 5\). We performed algebraic manipulation by first subtracting 2 from both sides. This simplified the problem to \(-3x \leq 3\).
Always perform the same operation on both sides of the inequality to maintain balance. In the next step, we divided both sides by -3. Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign. This step-by-step manipulation is key to correctly solving inequalities.
In our problem, we had the inequality \(2 - 3x \leq 5\). We performed algebraic manipulation by first subtracting 2 from both sides. This simplified the problem to \(-3x \leq 3\).
Always perform the same operation on both sides of the inequality to maintain balance. In the next step, we divided both sides by -3. Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign. This step-by-step manipulation is key to correctly solving inequalities.
Isolating Variables
Isolating variables is a crucial concept in solving linear inequalities. The goal is to get the variable alone on one side of the inequality, often making it easier to see the solution.
Let's delve into our problem. We started with \(2 - 3x \leq 5\). The first step was to isolate the variable term by removing the constant. We did this by subtracting 2 from both sides, resulting in \(-3x \leq 3\).
Next, to completely isolate \(x\), we divided by -3. This gave us \((x \geq -1)\), ensuring the variable was by itself, making the inequality easier to understand and interpret.
Remember: When isolating variables in inequalities, always perform identical operations on both sides and note the rule about reversing the inequality sign when dealing with negative coefficients.
Let's delve into our problem. We started with \(2 - 3x \leq 5\). The first step was to isolate the variable term by removing the constant. We did this by subtracting 2 from both sides, resulting in \(-3x \leq 3\).
Next, to completely isolate \(x\), we divided by -3. This gave us \((x \geq -1)\), ensuring the variable was by itself, making the inequality easier to understand and interpret.
Remember: When isolating variables in inequalities, always perform identical operations on both sides and note the rule about reversing the inequality sign when dealing with negative coefficients.
