Question

Solve the inequality. \(15 p<60 \quad\)

Short answer

p is less than 4, i.e., \( p < 4 \)

Step-by-step solution

Identify the inequality

The inequality present is \(15 p < 60\). It can be solved by isolating 'p'.

Solve the inequality

To isolate 'p', divide both sides by 15. Remember, dividing inequality by positive numbers doesn't change the direction of the inequality symbol. Thus, it becomes \( p < 60/15\).

Simplify the inequality

Divide 60 by 15, yields 4. Thus the final inequality is \( p < 4\).

Key concepts

Algebraic Inequalities

When it comes to understanding mathematical concepts, algebraic inequalities are among the foundational elements in algebra. An inequality compares two values, showing if one is less than, greater than, not equal to, or equal to another value but not strictly equal. Such expressions come in handy when solutions to problems are not fixed numbers but rather a range of possible solutions.

For example, the inequality given in the exercise, \(15 p < 60\), suggests that the product of 15 and some number \(p\) is less than 60. It's an open statement that might be true for many different values of \(p\). Solving inequalities involves finding all these possible values that satisfy the condition. The process of solving is similar to that of equations, with the addition of paying attention to how we manipulate the inequality to preserve its truth.

Isolate Variables

Isolating variables is a crucial skill when solving both equations and inequalities. To isolate the variable means to get the variable on one side of the equation or inequality, alone, and everything else on the other side. It involves performing operations such as addition, subtraction, multiplication, or division across the inequality, always mindful of keeping the balance.

In the step-by-step solution provided, isolating \(p\) was achieved by dividing both sides of the inequality by 15. A critical point to keep in mind is that when you multiply or divide both sides of an inequality by a negative number, the inequality symbol must be flipped. This rule ensures that the inequality's direction remains accurate. However, in this case, because we are dividing by a positive number, the direction remains the same.

Inequality Symbols

Inequality symbols are the mathematical symbols that represent the relationship between two expressions. The most common inequality symbols include < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Understanding the meaning of these symbols is essential for solving and graphing inequalities.

In our exercise, the key symbol is '<' which indicates that the value on the left (\(15p\)) is less than the value on the right (60). Remember that if you're asked to graph the solution on a number line, you'd draw an open circle at the number 4 and shade all the numbers to the left to represent all the numbers less than 4. It's important to remember that these symbols help define the range of solutions for an inequality, not a single solution as you would expect with an equation.