Problem 18 - Algebra 1: Concepts and Skills

Question

Describe the steps you would use to solve the inequality. $$ \frac{4}{3} t+5>\frac{1}{3} t $$

Step-by-Step Solution

Answer

The solution to the inequality is \(t > -5.\)

Step 1: Arrange Like Terms Together

Let's remove \(\frac{1}{3} t\) from the right-hand side to the left-hand side. By doing this we obtain: \(\frac{4}{3} t - \frac{1}{3} t + 5 > 0.\)

Step 2: Simplify the Expression

We simplify the expression by subtracting \(\frac{1}{3} t\) from \(\frac{4}{3} t\), therefore the inequality becomes: \(\frac{3}{3} t + 5 > 0.\)

Step 3: Isolate the Variable 't'

Let's isolate 't' by shifting '5' to the right-hand side, so we get \(t > -5.\)

Step 4: Final Concluding Step

The solution to this inequality is \(t > -5.\) That means the value of 't' in which the inequality holds true can be any real number greater than -5.

Key Concepts

Inequality Solutions

In mathematics, inequalities are statements that indicate the relative size or order of two values. When solving an inequality, like the sample exercise \(\frac{4}{3} t+5>\frac{1}{3} t\), the goal is to find all the possible solutions that make the inequality true. This typically involves finding a range of values for the variable in question. In our case, we are determining the set of all possible values of 't' that satisfy the given condition.

To effectively solve an inequality: first, collect all the variable terms on one side and the constant terms on the other, and then isolate the variable by performing permissible operations that maintain the inequality's direction. The final step generally involves expressing the solution in a simplified form, often as an inequality like \(t > -5\) or using interval notation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. In the given example, \(\frac{4}{3} t+5\) and \(\frac{1}{3} t\) are both algebraic expressions. Dealing with these expressions requires an understanding of how to manipulate them according to algebraic rules.

To simplify an algebraic expression, combine like terms (those terms that have the same variable raised to the same power), and perform operations within the restrictions set by the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). This systematic approach results in a less complex expression that is easier to work with when solving equations or inequalities.

Isolate Variables

Isolating the variable in an equation or inequality means rearranging the expression so that the variable is on one side of the equation or inequality sign by itself. This is a core technique used to solve for the variable. In the given problem, the variable 't' is isolated by first combining like terms and then moving the constant to the other side of the inequality.

When you isolate the variable, ensure to maintain balance on both sides of the equation or inequality. Any operation performed on one side (such as adding or subtracting a term, or multiplying or dividing by a number) must be performed on the other side as well. Following this step-by-step process is imperative to derive the correct solution.

Subtracting Fractions

Subtracting fractions is a basic mathematical skill that is widely used in algebra to combine terms with variables. To subtract fractions with like denominators, like \(\frac{4}{3}t - \frac{1}{3}t\) from our example, you simply subtract the numerators and keep the denominator the same. The process can involve reducing the resulting fraction to its simplest form by dividing the numerator and denominator by their greatest common factor.

For fractions with different denominators, you must first find a common denominator before you can subtract the numerators. In any case, the key to subtracting fractions is to ensure that the subtraction is done correctly to simplify the algebraic expressions and ultimately solve the equation or inequality at hand effectively.