Question

Find the solution sets of the given inequalities. $$ |4 x+5| \leq 10 $$

Short answer

The solution set is \([-\frac{15}{4}, \frac{5}{4}]\).

Step-by-step solution

Understanding the Absolute Inequality

The given inequality is \(|4x + 5| \leq 10\). This means that the expression inside the absolute value can be either between \(-10\) and \(10\). We need to consider two cases: when the expression \(4x + 5\) is less than or equal to \(10\) and greater than or equal to \(-10\).

Setting Up Two Inequalities

We start by setting up two inequalities without the absolute value: 1. \(4x + 5 \leq 10\)2. \(4x + 5 \geq -10\)These inequalities need to be solved individually to find the range of solutions for \(x\).

Solving First Inequality

Let's solve the first inequality, \(4x + 5 \leq 10\):Subtract 5 from both sides:\[ 4x \leq 5 \]Divide both sides by 4:\[ x \leq \frac{5}{4} \]

Solving Second Inequality

Now solve the second inequality, \(4x + 5 \geq -10\):Subtract 5 from both sides:\[ 4x \geq -15 \]Divide both sides by 4:\[ x \geq -\frac{15}{4} \]

Finding the Intersection

The solution set of the absolute inequality is the intersection of solutions from the two inequalities. The solution for \(x\leq \frac{5}{4}\) and \(x\geq -\frac{15}{4}\) can be combined as:\[ -\frac{15}{4} \leq x \leq \frac{5}{4} \]

Conclusion

Thus, the solution set is the interval \[ \left[-\frac{15}{4}, \frac{5}{4}\right] \]. This means that any value of \(x\) within this range satisfies the original inequality.

Key concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value symbols. The expression inside these symbols determines the range of values that satisfy the inequality. For the inequality \(|4x + 5| \leq 10\), it indicates that both positive and negative forms of the expression should fall within the bounds of -10 and 10.
This is because the absolute value symbol \(|...|\) represents the distance from zero. To solve these inequalities, you need to interpret them as two separate conditions.

• One where the expression is positive.
• Another where it is negative.

These transformations help convert the inequality into a solvable form without the absolute value. Understanding this concept is crucial for correctly solving absolute value inequalities.

Solution Sets

The solution set of an inequality is the collection of all possible values that satisfy the inequality. For absolute value inequalities, the solution set represents a range within which any chosen value will make the original inequality true.
In our example, after addressing both forms of the inequality, we found the solutions for each to be \(x \leq \frac{5}{4}\) and \(x \geq -\frac{15}{4}\).
To pinpoint the solution set, we merge these solutions.
This results in:

• \(-\frac{15}{4}\) as the lower limit.
• \(\frac{5}{4}\) as the upper limit.

Formally, it is expressed as \(\left[-\frac{15}{4}, \frac{5}{4}\right]\).
This interval encompasses all permissible values of \(x\) that make the inequality true, and they form the solution set.

Expression Solving

Breaking down and solving individual expressions is a fundamental part of finding the solution set. When the inequality was divided into two separate equations, the goal was to simplify each to find the value range for \(x\).
Solving \(4x + 5 \leq 10\):

• Subtract 5 from both sides to get \(4x \leq 5\).
• Divide each side by 4, resulting in \(x \leq \frac{5}{4}\).

Similarly, solving \(4x + 5 \geq -10\):

• Subtract 5 from both sides to derive \(4x \geq -15\).
• Divide each side by 4, giving \(x \geq -\frac{15}{4}\).

These steps outline the process of isolating \(x\) to identify the boundaries of its permissible values. It’s these actions that simplify the problem to reach a valid solution.

Inequality Intervals

When solutions from two separate inequalities come together, they form intervals on the number line. These intervals visually represent all numbers that meet the inequality's conditions.
By combining the results from both inequalities, an intersection is formed, representing overlapping solutions that work for \(|4x + 5| \leq 10\).
The results were \(x \geq -\frac{15}{4}\) and \(x \leq \frac{5}{4}\).
These intersect to form the interval:\[-\frac{15}{4}, \frac{5}{4}\].

• The square brackets show that the endpoints are included in the solution.
• Everything between and including these endpoints is a solution.

This concept elaborates on how multiple inequalities jointly cover a range, leading to intervals that display valid solutions. Understanding intervals is crucial when working with inequalities, as they simplify expressions into easy-to-interpret ranges.