Problem 65 - Introductory Algebra for College Students

Question

Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. \(5-x \leq 1\)

Step-by-Step Solution

Answer

\(x \geq 4\)

Step 1: Subtract from Both Sides

Start by subtracting 5 from both sides of the inequality to isolate 'x'. Thus, the inequality becomes: \(-x \leq -4\)

Step 2: Multiply by -1 and Switch Inequalities

After dividing both sides by -1, those sides become \(x \geq 4\). Remember that dividing or multiplying an inequality by a negative number switches the inequality sign.

Step 3: Graph the Solution

On a number line, you will shade in everything to the right of 4, including 4 itself, because our answer is \(x \geq 4\).

Key Concepts

Addition Property of Inequality

The Addition Property of Inequality is an essential principle used when solving inequalities. It states that you can add the same number to both sides of an inequality without changing the inequality. This property preserves the truth of the original inequality because the same value is being added to every part, ensuring balance.\\For example, consider an inequality like this: \(a < b\). If you add a number \(c\) to both sides, it becomes \(a + c < b + c\). The inequality remains true regardless of the value of \(c\).\\This concept helps maintain equilibrium while manipulating inequalities for solving. In the exercise provided, while subtraction of the same amount was applied initially, reversing the idea is also applicable through addition when isolating the variable.

Multiplication Property of Inequality

The Multiplication Property of Inequality allows you to multiply or divide both sides of an inequality by a positive number without changing the inequality's direction. However, when you multiply or divide by a negative number, the inequality's direction flips.\\In the context of the provided solution, we begin with the inequality \(-x \leq -4\). To isolate \(x\), multiply both sides by \(-1\). Because you are multiplying by a negative number, the inequality sign flips, resulting in \(x \geq 4\).\\Key points to remember include:

• Multiplying or dividing by a positive keeps the inequality the same.
• Multiplying or dividing by a negative flips the inequality direction.

\By understanding these rules, solving inequalities becomes intuitive and manageable.

Graphing Inequalities

Graphing inequalities helps to visually represent all possible solutions. When graphing the solution to an inequality such as \(x \geq 4\), a number line is used.\\Follow these steps for graphing an inequality:

• Identify the critical point, in this case, 4.
• Since the inequality is \(\geq\), include the point by using a closed circle on the number line.
• Shade the portion of the number line that represents all solutions, in this case, all numbers greater than or equal to 4, by shading to the right of 4.

\Graphing provides a clear and straightforward way to see the span of solutions an inequality comprises.

Number Line

A number line is a visual tool used to display numbers linearly. It is fundamental for graphing inequalities as it provides a way to show where solutions lie concerning other numbers.\\When solving inequalities:

• Mark numbers at regular intervals, ensuring they are evenly spaced.
• Use circles to denote specific values of interest. A closed circle represents inclusion (i.e., includes the number), while an open circle means exclusion.
• Shade regions according to the inequality direction to display the solution set.

\In the example of \(x \geq 4\), use a number line to highlight values starting at 4 and extending indefinitely to the right. The number line serves as a simple representation for variables' possible values.