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Chapter 6: Problem 34

Solve each inequality and graph the solution set on a number line. \(-7 x \leq-21\)

Short Answer

Expert verified
The solution to the given inequality is \(x \geq 3\). This means that every number greater than or equal to 3 is a solution. The graph has a solid dot or bracket at 3 with a line extending to the right, towards positive infinity.

Step by step solution

01

Isolate the variable

To get the variable \(x\) by itself, divide the inequality \(-7x \leq -21\) through by \(-7\). However, remember that when you divide or multiply an inequality by a negative number, the direction of the inequality sign changes. So, it becomes \(x \geq 3\).
02

The solution

Now the solution is obtained, which is all real numbers \(x\) that are more than or equal to 3.
03

Prepare the number line for graphing

Create a number line, which is a horizontal line with a scale. Make sure to include the number 3 in your scale since it's a part of the solution.
04

Graph the solution on the number line

Mark 3 on the number line with a solid dot or bracket (because the solution is \(x \geq 3\), so 3 is included in the solution). Draw a line extending to the right of 3, towards positive infinity, to show that all numbers greater than 3 are also solutions.

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Key Concepts

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

Graphing Solutions
Graphing solutions to inequalities helps us visually understand the range of values that satisfy them. To graph an inequality like \(x \geq 3\), we start by marking the number 3 on the number line. This is where our solution set begins. Since our inequality includes 3 (indicated by the "greater than or equal to" sign), we use a solid dot at this point.

Next, since the inequality signifies that \(x\) can be any number greater than or equal to 3, we draw a line extending from this solid dot towards the right, signifying all values more than 3. This line goes towards positive infinity, showing that the solution includes an infinite set of numbers extending outwards from 3.
Number Line
A number line is a simple yet powerful tool to represent numbers visually. It is a horizontal line where numbers are placed at equal intervals, providing an easy way to display solutions to equations and inequalities.

In solving our inequality, we plot our primary number, which is 3, on this line. We can use marks at regular intervals to indicate other numbers, ensuring that our primary number is clearly labeled. By visually marking numbers, it becomes easier to see the solution set and how it relates to other numbers. This visual aid makes understanding inequalities more intuitive.
Inequality Sign Change
When solving inequalities, an important rule to remember is that the inequality sign changes direction when you multiply or divide both sides by a negative number. This happens because the order of numbers on the number line reverses when multiplied by a negative.

For example, in the inequality \(-7x \leq -21\), dividing both sides by \(-7\) flips the inequality sign, changing \(\leq\) to \(\geq\). The new expression becomes \(x \geq 3\). Remembering this rule is crucial in maintaining the correct relationship between the numbers in your inequality solution.
Real Numbers
Real numbers include all the numbers you know: natural numbers, whole numbers, integers, rational numbers, and irrational numbers. They include both positive and negative numbers, fractions, and decimals.

In the context of our inequality \(x \geq 3\), the solution set includes any real number that is greater than or equal to 3. This means any decimal, fraction, or whole number starting from 3 and going upwards infinitely. Understanding real numbers helps in visualizing the expanse of solution sets in inequalities, showing that there are countless points on the number line that satisfy it.

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Most popular questions from this chapter

Factor the trinomials or state that the trinomial is prime. Check your factorization using FOIL multiplication. \(3 x^{2}+7 x+2\)

A new car worth \(\$ 24,000\) is depreciating in value by \(\$ 3000\) per year. After how many years will the car's value be \(\$ 9000 ?\)

Solve and check each equation. \(5 x-8=72\)

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