Problem 98 - Algebra and Trigonometry

Question

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. \(y=2 x-11+3(x+2)\) and \(y\) is at most 0

Step-by-Step Solution

Answer

The solution to the problem is \( x \in (-\infty, 1.4] \).

Step 1: Simplification of the expression

First, solve the expression \( y = 2x - 11 + 3(x+2) \) by distributing and combining like terms. This results in the simplified equation \( y = 5x -7 \), which is simpler to handle.

Step 2: Solve the inequality

Now we are given that \( y \leq 0 \). Subsitute y with the expression we found which is \( 5x -7 \). This gives us \( 5x -7 \leq 0 \). Now, solve this inequality for \( x \). Start by adding 7 on both sides of the inequality to isolate the \( x \) term. That results in \( 5x \leq 7 \). Finally, divide both sides by 5 and we get \( x \leq 1.4 \).

Step 3: Write the solution in interval notation

The solution to the inequality \( x \leq 1.4 \) is written in interval notation as \( (-\infty, 1.4] \). The square bracket indicates that the endpoint is included in the interval.

Key Concepts

Inequalities

An inequality is a mathematical statement that compares two expressions and shows the relationship between them. Inequalities use symbols like ">", "<", "≥", and "≤" to identify the type of relationship, such as greater than or less than.

• "Greater than" is represented by ">".
• "Less than or equal to" is represented by "≤".

Inequalities can be solved much the same way as equations; however, they have special rules, especially when dealing with negative numbers. For example, when both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed. In practical terms, inequalities are widely used to express ranges of possible values, like in our example with the condition that y is at most 0.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of one or more variables that make the equation true. This involves manipulating the equation using various algebraic operations to isolate the variable on one side. To achieve this:

• Add or subtract terms to move them from one side of the equation to the other.
• Use multiplication or division to get the variable by itself.

In our exercise, we solved the equation by distributing and combining like terms to simplify it. Then, we solved the inequality by isolating x to find the range of solutions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (addition, subtraction, multiplication, or division). They do not contain an equality sign, which distinguishes them from equations. Simplifying these expressions is an important step in solving algebraic problems, as seen in the exercise.
Simplification involves:

• Distributing numbers across parentheses using the distributive property.
• Combining like terms, which are terms that have the same variable raised to the same power.

For example, in the given expression for y, terms were distributed and combined to simplify it to something that is easier to manage and solve.

Linear Equations

Linear equations are a type of equation where the variable's highest power is 1. These equations form straight lines when graphed on a Cartesian coordinate system. The general form of a linear equation in one variable is ax + b = 0.
In the exercise, we dealt with a linear equation to find the value of x that satisfies the condition y ≤ 0. Solving linear equations is straightforward as it involves these basic steps:

• Isolate the variable on one side of the equation by using inverse operations.
• Simplify both sides as needed.
• Ensure that you perform the same operation on both sides of the equation to maintain balance.

These skills are crucial not only for solving equations themselves but also for understanding more complex relationships in algebra.