Problem 23 - Prealgebra

Question

Solve each inequality. Check your solution. Then graph the solution on a number line. $$-25 t \leq 400$$

Step-by-Step Solution

Answer

The solution is \(t \geq -16\).

Step 1: Isolate the Variable

To start solving the inequality \(-25t \leq 400\), we need to isolate the variable \(t\) by dividing both sides by \(-25\). Remember, dividing by a negative number reverses the inequality sign: \(t \geq \frac{400}{-25}\).

Step 2: Simplify the Division

Divide 400 by -25: \(\frac{400}{-25} = -16\). Thus, the inequality simplifies to \(t \geq -16\).

Step 3: Check the Solution

Choose a test value greater than or equal to \(-16\) to verify the inequality. Let's test \(t = -10\): \(-25(-10) \leq 400\right)\). Since \(250 \leq 400\) is true, our solution holds.

Step 4: Graph the Solution on a Number Line

Draw a number line and shade the region from \(-16\) to the positive side, indicating that \(t\) can be \(-16\) or any number greater. Place a closed dot on \(-16\) to include this value in the solution set.

Key Concepts

Solving Inequalities

In the realm of mathematics, solving inequalities is a fundamental skill that helps us understand relationships between numbers. An inequality is similar to an equation, but instead of stating that two expressions are equal, it tells us how one expression relates to another, using signs like ">", "<", "≤", or "≥". Here's how you solve an inequality:

• First, isolate the variable you're trying to solve for by performing equivalent mathematical operations on both sides of the inequality. This could involve adding, subtracting, multiplying, or dividing both sides by the same number.
• It's crucial to remember that when you multiply or divide both sides of an inequality by a negative number, the inequality sign reverses. This is because the direction of the inequality changes, altering the relationship between the expressions.
• In our exercise, we started with \(-25t \leq 400\). By dividing both sides by \(-25\), we got \(t \geq -16\). The sign flipped from less than or equal to, to greater than or equal to, because of the division by a negative.

Understanding these rules is the key to confidently solving inequalities. Practice regularly to become adept at handling different types of inequality problems.

Graphing Inequalities

Graphing inequalities helps visualize the solution set on a number line, providing a clearer picture of possible solutions. The graph of an inequality like \(t \geq -16\) shows all numbers that satisfy the inequality.

• Firstly, write down the endpoint of your inequality on the number line. For our example, \(-16\) is the point of interest.
• Decide if the endpoint should be included in the solution set. If the inequality sign is ">=" or "<=", the endpoint is included, depicted by a closed dot on the number line.
• If the inequality had been a strict inequality (">" or "<"), the endpoint would not be included, indicated by an open dot.
• Next, shade the region of the number line that represents the solution. For \(t \geq -16\), shade the line extending from \(-16\) to the right, indicating all numbers greater than or equal to \(-16\) are included in the solution.

Graphing not only affirms the solution but also offers a powerful way to understand how inequalities distribute numbers across the number line.

Number Line

A number line is an essential tool in mathematics, particularly when dealing with inequalities. It is a straight line where numbers are positioned at equal intervals, extending infinitely in both directions. This tool helps represent the set of solutions for inequalities visually.

• Each point on the number line represents a real number, and movements to the right signify increasing numbers, while movements to the left signify decreasing numbers.
• When graphing an inequality, use the number line to mark endpoint values and to shade the region representing the solution. This gives a visual cue of which numbers satisfy the inequality.
• In the example with the inequality \(t \geq -16\), a closed dot at \(-16\) and shaded line to the right clearly show that all numbers greater than or equal to \(-16\) are part of the solution.

The number line simplifies complex mathematical ideas by making abstract concepts more tangible and easier to manipulate visually. It's particularly useful when checking work for errors or when learning new concepts like inequalities.

Mathematical Operations

Mathematical operations, including addition, subtraction, multiplication, and division, are the backbone of solving inequalities. They allow manipulation of equations and inequalities to isolate variables and ultimately find their solutions.

• While handling equations, these operations follow straightforward logic: what's done to one side must be done to the other to keep balance.
• However, inequalities introduce a tiny twist. If you multiply or divide both sides by a negative number, the inequality flip-flops, swapping its direction. It means greater-than turns to less-than, or vice-versa.
• This fundamental rule makes sense—multiplying or dividing by a negative inherently changes the number's position relative to zero, switching its comparative state.

In our example, dividing \(-25t \leq 400\) by \(-25\) flipped the inequality sign to become \(t \geq -16\). Understanding and applying these operations properly ensures accurate solutions to inequalities.