Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 4 \geq 3 x+5>-1 $$

Step-by-Step Solution

Answer

The solution is \((-2, -\frac{1}{3}]\).

Step 1: Break Down the Compound Inequality

The given compound inequality is \(4 \geq 3x + 5 > -1\). We can split this into two inequalities: \(4 \geq 3x + 5\) and \(3x + 5 > -1\). We will solve these inequalities separately.

Step 2: Solve the First Inequality

For the inequality \(4 \geq 3x + 5\), we can solve it by isolating \(x\). Start by subtracting 5 from both sides, getting \(4 - 5 \geq 3x\), which simplifies to \(-1 \geq 3x\). Now, divide both sides by 3 to get \(-\frac{1}{3} \geq x\) or \(x \leq -\frac{1}{3}\).

Step 3: Solve the Second Inequality

For the inequality \(3x + 5 > -1\), subtract 5 from both sides to get \(3x > -1 - 5\), which simplifies to \(3x > -6\). Divide both sides by 3 to solve for \(x\), resulting in \(x > -2\).

Step 4: Combine the Solutions

Now, we combine the solutions from Step 2 and Step 3. We have \(x \leq -\frac{1}{3}\) from the first inequality and \(x > -2\) from the second. These can be combined into one interval. The solution is all \(x\) such that \(-2 < x \leq -\frac{1}{3}\).

Step 5: Express the Solution as an Interval

The solution \(-2 < x \leq -\frac{1}{3}\) can be expressed in interval notation. The interval is \((-2, -\frac{1}{3}]\), indicating all numbers greater than \(-2\) and less than or equal to \(-\frac{1}{3}\).

Key Concepts

Interval Notation

Interval notation is a mathematical notation used to describe a range of values. It offers a concise way to express solutions of inequalities. Here, brackets and parentheses are used to indicate the inclusion or exclusion of endpoints. In the final solution

• Square brackets, such as \([-\frac{1}{3}]\), indicate that the endpoint is included in the set.
• Parentheses, such as \((-2)\), indicate that the endpoint is not included in the set.

For our solution, we use both types of symbols, which results in the interval \((-2, -\frac{1}{3}]\). This means that \(x\) can be any value between \(-2\) and \(-\frac{1}{3}\), including \(-\frac{1}{3}\) but not including \(-2\). By employing interval notation, we can readily convey information about boundaries within the solution set.

Solving Inequalities

When solving inequalities, the goal is to determine the set of values that satisfy the inequality condition. Key steps include manipulating the expression to isolate the variable, similar to solving equations, but with additional rules:

• When multiplying or dividing both sides by a negative number, reverse the inequality sign.
• Perform arithmetic operations carefully, maintaining the direction of the inequality.

For example, in the inequality \(4 \geq 3x + 5\), we systematically cancel out numbers by subtracting 5 from both sides. Similarly, dividing by a positive 3 maintains the direction of the inequality, leading to \(x \leq -\frac{1}{3}\). These processes help us deduce a range of solutions adhering to the original condition.

Isolation of Variables

Isolation of variables is a technique used to solve equations or inequalities, enabling us to express the variable on one side of the inequality by itself. Through a sequence of operations, such as addition, subtraction, multiplication, or division, we strive to achieve this form.Consider the inequality \(3x + 5 > -1\). To isolate \(x\), we initially subtract 5 from each side, resulting in \(3x > -6\). Consequently, dividing by 3 isolates \(x\), resolving to \(x > -2\).

• The importance of this step lies in its ability to reveal the solution set that satisfies the condition.
• Ensuring variable isolation lays the groundwork for expressing the answer in interval notation and lends clarity to the range of valid values for \(x\).

By honing the skill of variable isolation, students simplify complex inequalities into understandable forms.