Problem 24 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(-2.5 x \leq-1.25\)

Step-by-Step Solution

Answer

The solution is \[ x \geq 0.5 \], or \[ [0.5, \infty) \] in interval notation.

Step 1 - Isolate the variable

To isolate the variable, divide both sides of the inequality by the coefficient of the variable. Given the inequality \(-2.5x \leq -1.25\), divide both sides by \(-2.5\).

Step 2 - Simplify the inequality

Remember, when dividing by a negative number, the inequality sign flips. \[\frac{-2.5x}{-2.5} \geq \frac{-1.25}{-2.5}\] This simplifies to \[x \geq 0.5\].

Step 3 - Graph the solution

To graph the inequality \(x \geq 0.5\), draw a number line. Place a solid dot (or closed circle) at \(0.5\) and shade the line to the right of \(0.5\) to indicate all values greater than or equal to \(0.5\).

Step 4 - Write the solution in interval notation

Since \(x \geq 0.5\), the interval notation is \[ [0.5, \infty) \]. This represents all numbers starting from \(0.5\) and increasing to positive infinity.

Key Concepts

solving inequalities

Solving inequalities is similar to solving equations but with a few extra rules. You aim to isolate the variable on one side. In the example \(-2.5 x \leq -1.25\), the first step is to divide both sides by \(-2.5\). This isolates \(x\). However, when you divide or multiply by a negative number, remember to flip the inequality sign. So, from \(-2.5 x \leq -1.25\), we end up with \(x \geq 0.5\). This technique is crucial for accurately solving inequalities.

graphing inequalities

After solving an inequality, you should graph it to visualize the solution set. For \(x \geq 0.5\), you'll draw a number line:

• Firstly, place a solid dot (or closed circle) at \(0.5\) because the inequality is 'greater than or equal to'.
• Then, shade the line to the right of this dot. This shading indicates all numbers that are greater than or equal to \(0.5\).

Graphing helps see all potential values that satisfy the inequality.

interval notation

Interval notation is a concise way to express the set of solutions to an inequality. For \(x \geq 0.5\), the interval notation is \[ [0.5, \infty) \].

• The square bracket \( [ \) indicates that \(0.5\) is included in the solution set (since the inequality is \('greater than or equal to')\).
• The parenthesis \( )\) near \( \infty \) shows that the set continues indefinitely towards positive infinity.

This notation efficiently describes all the numbers starting from \(0.5\) and going up to \(\text{infinity}\).