Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. $$ |1-4 x|-7<-2 $$

Short answer

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

Step-by-step solution

- Isolate the Absolute Value Expression

Start by isolating the absolute value expression. First, add 7 to both sides of the inequality:\[|1 - 4x| - 7 + 7 < -2 + 7\]This simplifies to:\[|1 - 4x| < 5\]

- Split into Two Inequalities

Since \( |1-4x| < 5 \), split this into two separate inequalities:\[1 - 4x < 5\]\[1 - 4x > -5\]

- Solve the First Inequality

Solve the first inequality \(1 - 4x < 5\) by isolating \(x\):\[1 - 4x < 5\]Subtract 1 from both sides:\[-4x < 4\]Divide by -4 (and reverse the inequality sign):\[x > -1\]

- Solve the Second Inequality

Solve the second inequality \(1 - 4x > -5\) by isolating \(x\):\[1 - 4x > -5\]Subtract 1 from both sides:\[-4x > -6\]Divide by -4 (and reverse the inequality sign):\[x < \frac{3}{2}\]

- Combine the Solutions

Combine the results from the two inequalities to find the solution set. The combined solution is:\[ -1 < x < \frac{3}{2} \]

- Express in Interval Notation

Express the solution set in interval notation:\[(-1, \frac{3}{2})\]

- Graph the Solution

Graph the solution on a number line. Mark open circles at -1 and at \(\frac{3}{2}\), and shade the region between them to represent all values between \( -1\) and \(\frac{3}{2}\):\[ \bullet ...................................\circ -1 \frac{3}{2} \]

Key concepts

absolute value inequalities

Absolute value inequalities involve expressions within absolute value bars, such as \(|a| < b\). To solve these inequalities, we need to understand the concept of absolute value. Absolute value represents the distance of a number from zero on a number line, without considering the direction. For example, both -3 and 3 have the same absolute value, which is 3. This property of absolute values makes solving the inequalities slightly different than regular inequalities.
When we have an inequality involving absolute values, like \[|1 - 4x| < 5\], it means that the expression inside the absolute value, \(1 - 4x\), must lie within 5 units of zero. This can be split into two separate inequalities: \[1 - 4x < 5\] and \[1 - 4x > -5\].
By solving these two inequalities, we can determine the range of values for \(x\) that satisfy the original inequality.

interval notation

Interval notation is a way of writing subsets of the real number line. It's particularly useful for expressing the solution sets of inequalities. In interval notation, we use brackets and parentheses to describe which parts of the number line are included in the solution set.
\(\) indicates that the endpoint is not included, while \[\] shows that the endpoint is included. For example, the solution set for an inequality like \(-1 < x < \frac{3}{2}\) can be written in interval notation as \((-1, \frac{3}{2})\). Here, the interval starts at -1 and ends at \(\frac{3}{2}\), but neither endpoint is included in the solution set.
Using interval notation helps in clearly and concisely expressing the range of values that satisfy the inequality.

solution sets

The solution set of an inequality is the set of all values that satisfy the inequality. To determine this, we solve the inequality step by step and combine the results.
For instance, from the given problem, after isolating the absolute value expression and splitting it into two inequalities \[1 - 4x < 5\] and \[1 - 4x > -5\], we solve each inequality separately:
\(1 - 4x < 5\)
Subtract 1 from both sides: \(-4x < 4\)
Divide by -4 and reverse the inequality sign: \(x > -1\)
Similarly, solve the other inequality:
\(1 - 4x > -5\)
Subtract 1 from both sides: \(-4x > -6\)
Divide by -4 and reverse the inequality sign: \(x < \frac{3}{2}\).
Combining these results, we get the solution set \(-1 < x < \frac{3}{2}\), which means all values of \(x\) between -1 and \(\frac{3}{2}\) are solutions to the given inequality.

graphical representation of inequalities

Graphical representation helps visualize the solution of inequalities on a number line. To graph the solution set \(-1 < x < \frac{3}{2}\), we follow these steps:

• Draw a number line.
• Mark the points -1 and \(\frac{3}{2}\) on it.
• Use open circles at these points because the inequality does not include the endpoints.
• Shade the region between -1 and \(\frac{3}{2}\) to represent all values of \(x\) that lie between these numbers.

This visual method helps confirm the range of solutions and makes it easier to understand the coverage of the solution set on the number line.