Problem 34 - Algebra 1: Concepts and Skills

Question

Check to see if x = 5 is or is not a solution of the equation or the inequality. $$ 16 \leq 3 x+1 $$

Step-by-Step Solution

Answer

Yes, \(x = 5\) is a solution of the inequality \(16 \leq 3x + 1\).

Step 1: Substitute the given value for x

Substitute \(x=5\) into the inequality. This changes \(16 \leq 3x + 1\) to \(16 \leq 3*5 + 1\).

Step 2: Simplify the right side of the inequality

Now we can carry out the calculation on the right side of the inequality. This changes \(16 \leq 3*5 + 1\) to \(16 \leq 15 + 1\).

Step 3: Check the inequality

Finally, check if the inequality \(16 \leq 15 + 1\) holds. It can be simplified to \(16 \leq 16\), which is a valid statement.

Key Concepts

Substitution Method

The substitution method is an essential technique used to evaluate the validity of a potential solution for an equation or inequality.

In practice, if you're given a value for a variable, like we see with the value of 5 for x in the example, you simply replace, or 'substitute', the variable with the provided number. Here's how it works in a systematic approach:

• Identify the variable within the inequality and its proposed solution. In this case, the variable is x, and the proposed solution is 5.
• Replace every instance of the variable in the inequality with the proposed solution. For the given inequality, we would substitute x with 5 to get \(3*5 + 1\).
• Evaluate the inequality to see if it holds true. As detailed in our step-by-step solution, after substitution, you assess whether \(16 \leq 15 + 1\) which indeed is a true statement, verifying the solution.

If the inequality remains true after the substitution, then the proposed value is indeed a solution. If it does not, the value is not a solution.

Inequality Solutions

Inequality solutions refer to the values that satisfy the inequality equation.

Unlike equalities, which typically have a single solution, inequalities can often have a range of solutions forming an interval. These solutions comprise all possible values that would make the inequality true. For instance, if the inequality was \(x < 5\), any value of x less than 5 would be considered a solution.

To check if a particular value is a solution, like x=5 in the original exercise, you insert that value into the inequality, which is what was done in the step-by-step solution. If the resulting statement is true, then that value is indeed a solution to the inequality. In our exercise, the inequality holds true when x=5, thus confirming that x=5 is a solution to the inequality \(16 \leq 3x + 1\).

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form while keeping their values unchanged. It's a key component in solving both equations and inequalities.

This process often includes:

• Combining like terms (e.g., \(2x + 3x\) simplifies to \(5x\))
• Performing arithmetic operations (like addition, subtraction, multiplying, and dividing)
• Applying distributive properties as necessary (e.g., \(a(b + c)\) simplifies to \(ab + ac\))

During the simplification in our inequality example, you saw how applying arithmetic operations (multiplying 3 by 5 and then adding 1) transforms \(3*5 + 1\) into \(15 + 1\), and further simplification brings you to the statement \(16\). This kind of simplifying helps in quickly evaluating whether the inequality is true or not. In the given solution, it confirmed that x = 5 is indeed a solution.