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Chapter 2: Problem 35

Solve each inequality and graph the solution on the number line. $$-3 x-7 > -1$$

Short Answer

Expert verified
The solution to the inequality is \(x < -2\), represented by an open circle at '-2' and a line extending to the left on the number line.

Step by step solution

01

Simplify the Inequality

Add 7 to both sides to remove the '-7' on the left side of the inequality. This gives you \(-3x > 6\).
02

Isolate 'x'

Divide every term by '-3' to isolate 'x'. Because we are dividing by a negative number, the direction of the inequality changes. This gives \(x < -2\).
03

Graphing the Solution

On the number line, 'x' is less than '-2', we represent the solution by drawing an open circle at '-2' (since '-2' is not included in the solution) and then drawing a line to the left.

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Key Concepts

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

Graphing Inequalities
When it comes to understanding inequalities, visual representation is an important aspect. Graphing inequalities allows you to see which numbers satisfy the inequality at a glance. Let's break it down step by step.
First, you need to solve the inequality algebraically. As shown in the example, you start by simplifying the inequality to isolate the variable.
Once your inequality is solved, you can graph it on the number line. Here's how:
  • Identify the boundary number (in this example, '-2').
  • Determine if the boundary is included in the solution. Here, the inequality is '<' meaning '-2' is not included, so we represent it with an open circle.
  • Shade or draw a line towards the values that satisfy the inequality ('x' less than '-2' means shading everything to the left of '-2').
This visual method enhances your understanding by clearly showcasing the range of values that make the inequality true.
Number Line Representation
A number line is a powerful tool for representing inequalities. It provides a simple way to illustrate where values fall in relation to each other. Let's see how this works for inequalities.
To represent an inequality on a number line, follow these steps:
  • Draw a horizontal line and mark it with sequential numbers at regular intervals.
  • Place an open or closed circle at the point that corresponds with the boundary number (using open for non-inclusive and closed for inclusive boundaries).
  • Shade or draw a line from the circle in the direction of the numbers that satisfy the inequality.
In our example, 'x < -2', the open circle at '-2' and shading to the left shows that any number less than '-2' is a solution.
Using a number line helps you quickly grasp which values make an inequality true, reinforcing your understanding of how inequalities function in a visually intuitive way.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations that define a specific relationship. To solve inequalities, understanding and manipulating algebraic expressions is crucial.
In the given example, the inequality starts as \[-3x - 7 > -1\] which is an algebraic expression. The following steps usually apply:
  • Simplify the expression by performing operations such as addition or subtraction to isolate terms.
  • Handle variables by dividing or multiplying, keeping in mind that dividing/multiplying by a negative reverses the inequality (< or >).
  • Check your final inequality for any errors in simplification or inequality direction.
Understanding algebraic expressions ensures that you approach inequalities methodically, solving them accurately while preserving the inequality's direction and integrity.

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