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Magazine Chapter 2: Problem 29
Solve each compound inequality. See Examples 4 and \(5 .\) $$ 1 \leq \frac{2}{3} x+3 \leq 4 $$
Short Answer
Expert verified
The solution is \(-3 \leq x \leq 1.5\).
Step by step solution
01
Understand the Compound Inequality
The inequality is written as \(1 \leq \frac{2}{3} x + 3 \leq 4\). This means that the expression \(\frac{2}{3} x + 3\) is greater than or equal to 1 and less than or equal to 4 simultaneously. We will solve it by splitting it into two separate inequalities.
02
Split the Inequality
We split the compound inequality into two separate inequalities:1. \(1 \leq \frac{2}{3} x + 3\)2. \(\frac{2}{3} x + 3 \leq 4\)We will solve these inequalities individually and then find the common value of \(x\) that satisfies both.
03
Solve the First Inequality
Solve the inequality \(1 \leq \frac{2}{3} x + 3\).- Subtract 3 from both sides: \(1 - 3 \leq \frac{2}{3} x\).- Simplify: \(-2 \leq \frac{2}{3} x\).- Multiply both sides by \(\frac{3}{2}\) to isolate \(x\): \(-3 \leq x\).This indicates that \(x \geq -3\).
04
Solve the Second Inequality
Now solve the inequality \(\frac{2}{3} x + 3 \leq 4\).- Subtract 3 from both sides: \(\frac{2}{3} x \leq 1\).- Multiply both sides by \(\frac{3}{2}\) to isolate \(x\): \(x \leq \frac{3}{2}\).This indicates that \(x \leq 1.5\).
05
Combine the Solutions
Now we combine the solutions from the two separate inequalities. We have:- From Step 3: \(x \geq -3\).- From Step 4: \(x \leq 1.5\).The values that satisfy both inequalities are \(-3 \leq x \leq 1.5\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inequality Solving
Solving inequalities is a vital skill in algebra, as it allows us to find a range of possible values for variables instead of a single solution. Inequalities show that one quantity is greater than or less than another. Unlike equations, which use equal signs, inequalities use signs like ">", "<", ""). Here's how it works:
- Identify the inequality: Recognize whether it's a single inequality or a compound one that combines multiple conditions.
- Isolate the variable: Apply operations to both sides to solve for the variable. Keep track of operations to maintain the inequality's direction.
- Checking solutions: Verify the solution by substituting values into the original inequality.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent a particular value or set of values. They form the building blocks of algebra that allow us to model and solve real-world situations.
- Components: Consist of terms, which can be constants (numbers on their own) or variables (letters representing numbers).
- Operations: Include addition, subtraction, multiplication, and division, which act on the terms.
- Simplification: Involves combining like terms and performing arithmetic operations to reduce expressions to their simplest form.
Linear Inequalities
Linear inequalities resemble linear equations but use inequality symbols (such as "<" and ">") instead of equal signs. They graphically represent a range of solutions rather than a single point, often forming a half-plane on a graph.
- Form: A typical form is \(ax + b \leq c\), where \(a, b, \) and \(c\) are constants.
- Solution methods: Isolate the variable using inverse operations (addition, subtraction, multiplication, and division). When multiplying or dividing by a negative number, reverse the inequality sign.
- Graphing: Solutions appear as shaded regions on a number line or coordinate plane, illustrating all possible solutions.
