Problem 76 - Introductory and Intermediate Algebra for College Students 4th

Question

Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. $$7-2(x-4)<5(1-2 x)$$

Step-by-Step Solution

Answer

In short, the solution for given inequality is \( x<-5/6 \), and its solution set can be represented on the number line with a ray starting from \( -5/6 \) and extending to the left excluding \( -5/6 \).

Step 1: Expand the inequality

Begin by expanding the inequality by distributing \( -2 \) to \( (x-4) \) and \( 5 \) to \( (1-2x) \). It gives us the inequality, \( 7-2x+8<5-10x \).

Step 2: Simplify the inequality

Combine like terms on each side of the inequality which leads to \( 15+2x<5-10x \).

Step 3: Isolate the term with x on one side of the inequality

To do this, add \( 10x \) to both sides of the equation giving us \( 15+12x<5 \).

Step 4: Further Isolate x

Subtract \( 15 \) from both side leads to \( 12x<-10 \).

Step 5: Solve for x

Finally, divide both sides by \( 12 \) to find the value of \( x \). This gives \( x<-10/12 \) or \( x<-5/6 \).

Step 6: Graph the solution set

On a number line, we represent all the numbers less than \( -5/6 \). The solution will be a ray that starts at point \( -5/6 \) and extends to the left. The point \( -5/6 \) is not included in the solution set, so we use an open circle.

Key Concepts

Addition Property of Inequality

Inequalities are quite similar to equations, and you often manipulate them using rules that closely resemble arithmetic operations. The addition property of inequality is one such rule. This principle states that if you add or subtract the same number from both sides of an inequality, the inequality's direction does not change. For example, if you have an inequality: - \( a < b \) Then by adding \( c \) to both sides, the inequality remains: - \( a + c < b + c \) This property is particularly useful when you want to isolate a variable on one side of an inequality. You can move a term from one side to the other by adding or subtracting it, just as you would in an equation.

• Important: When shifting terms, be sure to perform the same operation on both sides of the inequality.

In our exercise, we use the addition property by adding \( 10x \) to both sides in step 3 to rearrange the inequality and begin isolating \( x \). Understand that this step does not alter the inequality's direction, keeping it unbeaten and reliable.

Multiplication Property of Inequality

Multiplication is another crucial operation when dealing with inequalities. The multiplication property of inequality states that the inequality's direction remains unchanged when you multiply or divide both sides by a positive number. However, when you multiply or divide both sides by a negative number, you must reverse the inequality's direction to maintain its logic. Here's a basic breakdown:

• If \( a < b \) and you multiply by a positive \( c \) (where \( c > 0 \)), then \( ac < bc \).
• But if you multiply by a negative \( c \) (where \( c < 0 \)), the inequality reverses, so \( ac > bc \).

In the given solution, step 5 involves dividing by 12. Since 12 is positive, we don't flip the inequality sign while solving for \( x \). Understanding when to flip the sign is essential for correctly solving inequalities, so always pay attention to the signs of numbers involved.

Graphing Inequalities

Visualizing inequalities can greatly aid understanding, and graphing them is a common method for this. Graphing begins by solving the inequality algebraically, finding values that satisfy it. Once you've isolated the variable (like \( x \) in our exercise), you can represent this solution using a graph on a number line.

• For instance, if \( x < -\frac{5}{6} \): this means \( x \) can take any value smaller than \( -\frac{5}{6} \).
• If the inequality were \( \leq \) or \( \geq \), the point would include a filled circle, indicating the endpoint is part of the solution.

Graphing inequalities is quite helpful because it gives a visual representation of the infinite set of solutions the inequality encompasses. You can see at a glance which values make the inequality true and better understand the problem's domain.

Number Line Representation

A number line is a simple yet powerful visual tool for representing solutions of inequalities. Using a number line, you can precisely show ranges of numbers that satisfy an inequality. Here's the step-by-step of how we represent our exercise's solution:

• Identify the key point, which is \( -\frac{5}{6} \).
• Use an open circle at \( -\frac{5}{6} \) since numbers strictly less than this point are included in the solution.
• Draw a ray extending leftwards from the open circle to indicate that every number less than \( -\frac{5}{6} \) meets the inequality's requirement.

Number lines are excellent for showing the range and scope of solutions clearly and immediately. An open circle means the endpoint is not included, which directly relates back to the original inequality situation resolved algebraically. Just remember: keep it clear and straightforward—your visual representation should always mirror your algebraic solution.