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Magazine Chapter 1: Problem 21
Solve the inequality. $$ -1 \leq 2 x-3<4 $$
Short Answer
Expert verified
The solution is \(1 \leq x < \frac{7}{2}\).
Step by step solution
01
Split the Compound Inequality
The given inequality is compound, consisting of two inequalities combined. We need to solve each part separately. So let's first break it down into two inequalities:1. \(-1 \leq 2x - 3\)2. \(2x - 3 < 4\)
02
Solve the First Inequality
We begin with the inequality \(-1 \leq 2x - 3\). To solve it, we first add 3 to both sides:\(-1 + 3 \leq 2x\)\(2 \leq 2x\)Next, divide both sides by 2:\(\frac{2}{2} \leq \frac{2x}{2}\)\(1 \leq x\)
03
Solve the Second Inequality
Now let's solve the second inequality: \(2x - 3 < 4\). Add 3 to both sides to isolate the \(2x\):\(2x - 3 + 3 < 4 + 3\)\(2x < 7\)Now, divide both sides by 2:\(\frac{2x}{2} < \frac{7}{2}\)\(x < \frac{7}{2}\)
04
Combine the Solutions
Now that we have solutions for both inequalities, we combine them. From Step 2, we have \(1 \leq x\) and from Step 3, we have \(x < \frac{7}{2}\). Combining these two results gives us:\(1 \leq x < \frac{7}{2}\).Therefore, the solution to the compound inequality is that \(x\) lies between 1 and \(\frac{7}{2}\), inclusive of 1 and exclusive of \(\frac{7}{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Compound Inequalities
A compound inequality involves two separate inequalities that are connected through either "and" or "or". In this case, the given compound inequality, \(-1 \leq 2x - 3 < 4\), uses the "and" conjunction implicitly. This means we are looking for values of \(x\) that satisfy both parts of the inequality simultaneously.
To approach a compound inequality:
Understanding how to manage compound inequalities is essential, as it will help in finding ranges of solutions and understanding where these solutions apply in real-life scenarios.
To approach a compound inequality:
- Break it into two separate inequalities.
- Solve each one individually.
- Combine the solutions according to the logical operator that connects them. For an "and", both conditions must be true. In an "or", at least one condition must be true.
Understanding how to manage compound inequalities is essential, as it will help in finding ranges of solutions and understanding where these solutions apply in real-life scenarios.
Inequality Solution
Solving an inequality means finding all possible values of the variable that make the inequality true. The solution forms a range of values rather than a specific number. When dealing with inequalities, remember:
In our exercise, the solution is expressed in interval notation: \(1 \leq x < \frac{7}{2}\). This tells us that \(x\) can be from 1 up to but not including \(\frac{7}{2}\).
- The direction of the inequality sign is crucial. It shows whether the values for \(x\) are lesser or greater.
- When multiplying or dividing by a negative number, the inequality direction must be flipped. However, in the given exercise, this rule isn鈥檛 applied as all operations involve positive numbers.
- Greater rigour is required as the solution can include or exclude endpoints, indicated by open (<) or closed (鈮�) intervals.
In our exercise, the solution is expressed in interval notation: \(1 \leq x < \frac{7}{2}\). This tells us that \(x\) can be from 1 up to but not including \(\frac{7}{2}\).
Mathematical Steps
Mathematical steps are logical sequences we follow to arrive at a solution. In solving inequalities, each step must be performed correctly to ensure the solution is valid. Let's go through the steps applied in the solution:
Following these mathematical steps establishes a clear and structured path to achieving a correct and comprehensive inequality solution.
- First, address each part of the compound inequality separately. Split it into two inequalities: \(-1 \leq 2x - 3\) and \(2x - 3 < 4\).
- For each inequality, isolate the variable by performing operations such as addition, subtraction, multiplication, or division. Carefully track each step to avoid altering the inequality's meaning.
- After solving both inequalities, overlap the solutions to achieve the overall solution for the compound inequality. This ensures that only the values satisfying both conditions are included in the final answer.
Following these mathematical steps establishes a clear and structured path to achieving a correct and comprehensive inequality solution.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations or inequalities to make them easier to solve. In inequalities, often it's about getting the variable on one side of the inequality sign. Key actions in algebraic manipulation include:
In our example:
- Adding or subtracting terms from both sides of an inequality without changing the inequality direction.
- Dividing or multiplying by a positive number, which also preserves the inequality direction.
- Being cautious with negative numbers鈥攊f you need to multiply or divide by a negative, you must switch the direction of the inequality.
In our example:
- Both inequalities required straightforward manipulation: add or subtract to isolate terms, then divide to solve for \(x\).
- No complex algebraic techniques were necessary, but precision is key.
- Properly handling these manipulations gives a precise solution range, teaching how to balance an equation intuitively and accurately.
