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Magazine Chapter 1: Problem 35
Solve the given inequality. Write the solution set using interval notation. Graph the solution set. $$ |-5 x|<4 $$
Short Answer
Expert verified
The solution set is \((-4/5, 4/5)\).
Step by step solution
01
Understand the Absolute Value Inequality
The absolute value inequality \(|-5x|<4\) asks for all values of \(x\) that make the absolute value of \(-5x\) less than 4. Since absolute value measures the distance from zero, we will split this into two separate inequalities.
02
Split the Inequality Into Two Cases
The inequality \(|-5x| < 4\) implies two conditions: \(-5x < 4\) and \(-5x > -4\), because absolute value less than a number means less than that number and greater than the opposite of that number.
03
Solve the First Inequality
For \(-5x < 4\), divide both sides by -5, remembering to flip the inequality sign. Thus, we get:\[x > -\frac{4}{5}\]
04
Solve the Second Inequality
For \(-5x > -4\), divide both sides by -5, remembering to flip the inequality sign. Thus, we get:\[x < \frac{4}{5}\]
05
Combine the Solution
The solutions to the inequalities \(x > -\frac{4}{5}\) and \(x < \frac{4}{5}\) suggest that \(x\) lies between these two values. Therefore, the solution set in interval notation is:\[(-\frac{4}{5}, \frac{4}{5})\]
06
Graph the Solution
On a number line, plot an open interval between \(-\frac{4}{5}\) and \(\frac{4}{5}\). This visual representation shows all possible values of \(x\) satisfying the inequality \(|-5x| < 4\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation is a symbolic way of expressing a range of values. It is especially useful in math when dealing with inequalities. When we write an interval, we're essentially describing a set of numbers that satisfy a given condition. For example, in the inequality \(-\frac{4}{5} < x < \frac{4}{5}\), we're looking for all the numbers between \(-\frac{4}{5}\) and \(\frac{4}{5}\), but not including these endpoints. This is described in interval notation as \((-\frac{4}{5}, \frac{4}{5})\).
- The parentheses \((\) and \()\) mean simply less than (<) or greater than (>), indicating that the endpoints are not included.
- If the inequality involves \(\leq\) or \(\geq\), we would use square brackets \([\) and \()]\), showing that the endpoints are included.
Inequality Solution
Solving inequalities involves finding values of the variable that make the inequality true. For absolute value inequalities like \(|-5x| < 4\), the method involves breaking it down into two separate inequalities. This is because the absolute value measures the distance from zero, meaning \(-5x\) can either be positive or negative.
- For the inequality \(-5x < 4\), by dividing both sides of the inequality by \(-5\), we need to flip the inequality sign, resulting in \(x > -\frac{4}{5}\).
- For the second inequality \(-5x > -4\), a similar operation gives us \(x < \frac{4}{5}\).
Graphing Inequalities
Graphing inequalities is a visual way to show the solution set on a number line. For the inequality \(|-5x| < 4\), the solution was \((-\frac{4}{5}, \frac{4}{5})\), which translates to specific guidelines for the graph.
- We draw an open circle at both \(-\frac{4}{5}\) and \(\frac{4}{5}\) to indicate that these endpoints are not included in the solution set.
- The area between these open circles is shaded, representing all the values of \(x\) that are between \(-\frac{4}{5}\) and \(\frac{4}{5}\).
