Problem 30 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. $$ -0.3 x>-2.4 $$

Step-by-Step Solution

Answer

The solution is \(x < 8\), represented by an open circle at 8 with shading to the left.

Step 1: Understanding the Inequality

We need to solve the inequality \(-0.3x > -2.4\). Our goal is to isolate \(x\) on one side to find its possible values.

Step 2: Isolate x by Dividing

To isolate \(x\), we need to divide both sides of the inequality by \(-0.3\). Remember that dividing by a negative number reverses the inequality sign:\[x < \frac{-2.4}{-0.3}\]

Step 3: Simplify the Quotient

Calculate \(\frac{-2.4}{-0.3}\):This simplifies to \(8\) because \(-2.4 \div -0.3 = 8\). Thus, our new inequality becomes:\[x < 8\]

Step 4: Graph the Solution

To graph \(x < 8\), draw a number line. Mark an open circle at \(8\) to show that \(8\) is not included in the solution. Shade everything to the left of \(8\) to represent numbers less than \(8\).

Key Concepts

Solving Inequalities

Solving inequalities is much like solving equations, but with some crucial differences. Here, our primary aim is to determine the range of values that satisfy the inequality. Let's consider the inequality \(-0.3x > -2.4\). To find the solution, our goal is to isolate the variable, \(x\).

• First, identify the operation affecting \(x\), which in this case is multiplication by \(-0.3\).
• To solve for \(x\), we perform the inverse operation: dividing both sides by \(-0.3\).

However, an important rule in inequalities states that when you divide or multiply both sides by a negative number, the inequality sign must be flipped. This step transforms our inequality into:\[x < \frac{-2.4}{-0.3}\]After performing the division, we find \(x < 8\). This tells us that all values less than 8 are solutions to the inequality.

Graphing Inequalities

Graphing inequalities is a visual way to showcase all possible solutions on a number line. Once we have the inequality \(x < 8\), we need to graph it to represent the solution set. Starting with a number line, you'll:

• Mark a point at \(x = 8\) to indicate the boundary.
• Use an open circle on \(8\), which shows that 8 is not included in the solutions as the inequality is a strict \(<\), meaning 'less than' without equality.
• Shade to the left of the open circle across the number line. This shaded region represents all numbers that are less than 8.

Graphing in this way provides a clear, visual understanding of where the solutions lie and how extensive they can be.

Negative Coefficients in Inequalities

Negative coefficients in inequalities require special attention. In our inequality, \(-0.3x > -2.4\), the coefficient of \(x\) is negative. Whenever you divide or multiply an inequality by a negative number, it alters the direction of the inequality sign.

• This reversal reflects a fundamental property of inequalities.
• It helps maintain truthfulness across the inequality. For instance, if \(a > b\), multiplying by \(-1\) would imply that \(-a < -b\).

In our example, dividing both sides by \(-0.3\) flips \(>\) to \(<\), resulting in \(x < 8\). Awareness of this rule is critical as overlooking it can lead to incorrect solutions.