Question

Solve the inequality. $$6 x+5<23$$

Short answer

The solution to the inequality is \(x < 3\).

Step-by-step solution

Subtract 5 from both sides

The first step is to isolate the term with the variable, in this case \(6x\), on one side of the inequality. To do this, subtract \(5\) from both sides of the inequality. Hence, \(6x + 5 - 5 < 23 - 5\), which simplifies to \(6x < 18\).

Divide by 6

Next, divide both sides of the inequality by \(6\) to solve for \(x\). Thus, \(x < 18 / 6\) simplifies to \(x < 3\).

Key concepts

Algebraic Expressions

Algebraic expressions are a fundamental aspect of algebra, consisting of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They represent values and allow us to describe mathematical relationships. In the context of inequality, as seen with the problem \(6x + 5 < 23\), algebraic expressions form one or both sides of the inequality sign.

To understand these expressions better, we first identify the terms involved. In our example, there are two terms on the left-hand side: \(6x\) and \(5\). The number before the variable \(x\) is known as the coefficient, and it signifies how much the variable is multiplied by. The term without a variable is a constant, which in this case adds a fixed value of \(5\) to the variable term. Understanding the role of each part of an algebraic expression is crucial in manipulating and solving inequalities.

Isolate Variables

Isolating variables is a key skill in solving algebraic inequalities and equations. The goal is to rearrange the inequality so that the variable is by itself on one side. This is achieved through strategic 'inverse operations.'

Look at the example \(6x + 5 < 23\). To isolate the variable \(x\), we perform operations that reverse the effect of whatever is currently being done to \(x\). Since \(x\) is being multiplied by \(6\) and \(5\) is added to it, we first subtract \(5\) from both sides to remove the constant term. This yields a simpler expression, \(6x < 18\), where \(x\) is only being multiplied. Next, we divide both sides by \(6\), the coefficient of \(x\), thus isolating the variable and finding the solution \(x < 3\). Inequalities like \(6x + 5 < 23\) can be thought of as a balance scale; whatever operation you perform on one side, you must also do to the other to maintain the balance.

Inequality Manipulation

Inequality manipulation involves applying mathematical operations to both sides of an inequality while preserving its direction unless specific circumstances arise, such as multiplying or dividing by a negative number. In our example, \(6x + 5 < 23\), the solution process requires careful manipulation of this inequality to solve for \(x\).

After isolating the variable, we get \(6x < 18\). Dividing both sides by \(6\), a positive number, keeps the direction of the inequality the same. Hence, the solution becomes \(x < 18 / 6\), so \(x < 3\). It’s important to note that if we had to divide by a negative number, the inequality sign would flip. Understanding these rules is essential to correctly manipulating inequalities without making errors. Always be aware of the signs involved, and remember to flip the inequality if you multiply or divide by a negative.