Problem 32 Sketch the graph of the inequali... [FREE SOLUTION]

Chapter 6: Problem 32

Sketch the graph of the inequality. $$-2 x \geq 10$$

Short Answer

Expert verified

The graph of the inequality $ -2x \geq 10 $ is a vertical line at $ x = -5 $, with all the area to the left of this line shaded to indicate the set of solutions.

Step by step solution

Simplify the Inequality for x

First, the inequality needs to be simplified to find the value for x. This involves dividing both sides of the inequality by -2 (which is the coefficient of x). It should be noted that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes. Thus, the inequality -2x >= 10 simplifies to x <= -5.

Plot the line x = -5

The inequality $x \leq -5$ represents all values of x that are less than or equal to -5. This can be represented graphically by a vertical line which passes through the point x = -5 on the x-axis. As equal to -5 is included (since it's $\leq -5$ and not $< -5$), the line is drawn as a filled line, not dotted.

Indicate the solution to the inequality on the graph

As the inequality $ x \leq -5 $ represents all values of $ x $ that are less than or equal to -5, it means all points on the left-hand side of the line $ x = -5 $ are included in the solution. This can be indicated on the graph by shading the area to the left of the line $ x = -5 $.

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Key Concepts

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

Solving Inequalities

Inequalities are statements about the relative size of two values. They use signs like <, >, 鈮�, and 鈮� to show how one expression compares to another. Solving inequalities is much like solving equations but with some key differences. Here鈥檚 a simple guide:

Start by simplifying the inequality to get the unknown variable by itself on one side. This typically involves adding, subtracting, multiplying, or dividing.
Be particularly cautious when multiplying or dividing both sides of the inequality by a negative number; this action requires you to flip the direction of the inequality sign. For example, if you multiply or divide by -2, a greater than sign (>) becomes a less than sign (鈮�), and vice versa.
The final result should give you a range or set of possible values that satisfy the inequality, often expressed in the form of an inequality like x 鈮� -5.

Understanding these steps is vital to solving inequalities effectively. It's all about figuring out which values make the inequality true.

Graphing Techniques

Graphing inequalities helps visualize their solutions and understand them better. Here's how you can graph an inequality like x 鈮� -5:

First, you start by plotting the line corresponding to the boundary of the inequality (x = -5 in this case). This line acts as a threshold that separates solutions from non-solutions.
Since x 鈮� -5 includes the value at x = -5 itself, the line is solid, indicating that this boundary is part of the solution.
Once the line is in place, shade the area where the inequality holds true. For the inequality x 鈮� -5, this means shading to the left of the line, as all these values are less than or equal to -5.

These simple plotting steps help in translating algebraic expressions into visual forms, making it easier to see and understand the range of solutions.

Algebraic Solutions

Using algebra to solve inequalities involves manipulating terms to isolate the variable, much like solving regular equations. But it's important to remember a few special rules:

When you require dividing or multiplying both sides by a negative number, don't forget to reverse the inequality sign.
If you're dealing with compound inequalities, such as those that involve both 'and' or 'or,' simplify each separate part before combining them into a final solution.
Compare these solutions by checking specific values back in the original inequality to ensure they satisfy all conditions.

Algebraic solutions are powerful because they give clear, precise boundaries for the set of numbers that satisfy the inequality. By following a step-by-step method, you can understand not only how to reach a solution, but also why it works this way, enhancing your mathematical reasoning.

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91影视

Sketch the graph of the inequality. $$-2 x \geq 10$$

Short Answer

Step by step solution

Simplify the Inequality for x

Plot the line x = -5

Indicate the solution to the inequality on the graph

Key Concepts

Solving Inequalities

Graphing Techniques

Algebraic Solutions

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