Problem 73 - Algebra 1

Question

Check whether the given number is a solution of the inequality. $$ 16 p-9 \leq 71 ; 5 $$

Step-by-Step Solution

Answer

Yes, the given number 5 is a solution to the inequality.

Step 1: Solve the Inequality for 'p'

The given inequality is \(16p - 9 \leq 71\). First, add 9 to both sides of the inequality to isolate the term with 'p'. This yields: \(16p \leq 71 + 9\), which simplifies to: \(16p \leq 80\). To find 'p', divide both sides by 16, resulting in the inequality: \(p \leq 5\).

Step 2: Test the Given Number

The number to be tested is 5. Substitute 5 for 'p' in the inequality \(p \leq 5\), resulting in: \(5 \leq 5\). Both sides of the inequality are equal, implying that 5 is indeed a solution to the inequality.

Key Concepts

Solving Inequalities

Solving inequalities is similar to solving equations, but with a twist. In inequalities, we are finding a range of possible values that satisfy the given condition, rather than a single solution. For example, when we have an inequality like \(16p - 9 \leq 71\), we're looking for all possible values of \(p\) that make this statement true.
To solve it effectively, you need to:

• First, perform operations to isolate the variable on one side of the inequality.
• Remember that adding or subtracting the same number from both sides doesn’t change the inequality.
• When you multiply or divide by a negative number, the direction of the inequality changes.

In our exercise, we kept the direction of the inequality when we added and divided by positive numbers. This resulted in finding that \(p \leq 5\), which means \(p\) can be any number less than or equal to 5.

Substitution Method

The substitution method is a very effective way to check solutions for inequalities and equations. This method requires that you substitute the given value into the inequality to see if it makes the statement true or not.
Here's the step-by-step process:

• Take the number you need to test — in this case, 5.
• Substitute this number for the variable in the inequality or equation.
• Calculate to see if the resulting statement is true or false.

In our example, substituting \(p = 5\) into the inequality \(5 \leq 5\) resulted in a true statement because both sides are equal, confirming that 5 is a valid solution to the inequality \(16p - 9 \leq 71\). Using substitution gives a straightforward way to validate solutions.

Algebraic Solutions

Finding algebraic solutions involves manipulating expressions to isolate variables and find conditions that satisfy the given equation or inequality. It involves engaging with algebraic techniques such as:

• Combining like terms.
• Using inverse operations to isolate variables.
• Ensuring each step of transformation maintains equality or the inequality's order intact.

For the inequality \(16p - 9 \leq 71\), we added 9 to counteract the negative 9 and then divided by 16 to solve for \(p\). The solution \(p \leq 5\) was reached without disturbing the inequality's balance.
Algebraic solutions offer a precise approach to determining solutions, using arithmetic and logical steps. This method helps in understanding how inequalities differ from equations in that solutions often involve ranges of values.