Problem 35 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(0.02 x+5.58<0\)

Step-by-Step Solution

Answer

The solution is \(x < -279\).

Step 1: Isolate the variable term

To solve the inequality, first isolate the term containing the variable. Subtract 5.58 from both sides of the inequality: \[ 0.02x + 5.58 - 5.58 < 0 - 5.58 \]This simplifies to: \[ 0.02x < -5.58 \]

Step 2: Solve for the variable

Next, solve for \(x\) by dividing both sides of the inequality by 0.02:\[ \frac{0.02x}{0.02} < \frac{-5.58}{0.02} \]This simplifies to:\[ x < -279 \]

Step 3: Graph the solution set on a number line

To graph the solution \(x < -279\), draw a number line. Place an open circle at -279 to indicate that -279 is not included in the solution. Shade the line to the left of -279 to indicate that all values less than -279 are included.

Key Concepts

Graphing Inequalities

Graphing inequalities is an essential part of understanding how inequalities work and interpreting their solutions. When we solve an inequality algebraically, we're typically left with an expression that shows us which values satisfy the condition of the inequality. To graph this on a number line, follow these simple steps:

• Find the boundary point(s) which are usually found by considering equality. For example, if you have the inequality like the one in our problem, " x < -279 ", -279 is your boundary point.
• Determine whether the boundary point is included or not. If the inequality is "< " or ">", use an open circle to show the number is not part of the solution. If it is "≤" or "≥", use a closed circle.
• Shade in the number line to indicate the range of solutions. For "x < -279", you would shade everything to the left of -279.

This visual representation helps you quickly see all possible solutions for the inequality.

Number Line

A number line is a simple, yet powerful tool for visualizing and understanding inequalities. It is a visual representation of numbers where each point on the line corresponds to a number. Here's how you can utilize a number line effectively for inequalities:

• Start by drawing a horizontal line. Make sure it's long enough to mark the numbers you need to represent.
• Label key points that are relevant to the problem, such as -279 in our inequality example.
• Use symbols like open or closed circles to indicate whether endpoints are included in the inequality.

This method allows you to see clearly which numbers fall within the solution set of the inequality. It makes it immediately apparent if a number is a solution of the inequality or not.

Isolate Variable

Isolating the variable is a crucial first step when solving inequalities, as it helps simplify the inequality and makes it easier to understand what value of the variable will satisfy it. Let's break down this process:

• Start with the original inequality. In our case, it's "0.02x + 5.58 < 0." The goal is to have the variable, here "x", on one side of the inequality by itself.
• Use basic algebraic operations such as addition, subtraction, multiplication, or division to move other terms to the opposite side of the inequality. Here, we first subtracted 5.58 from both sides.
• Once you have isolated the variable term — in our example it simplifies to "0.02x < -5.58" — proceed by clearing any coefficients or factors. Divide by 0.02 to solve for "x."

This systematic approach helps simplify complex inequalities, making them more manageable, and allows you to graph and further analyze them effectively.