Absolute Value Inequalities: Videos & Practice Problems

Solving absolute value inequalities where the absolute value expression is less than or less than or equal to a number involves combining techniques from solving absolute value equations and inequalities. When faced with an inequality such as |x| < a, it means the distance between x and zero is less than a. This can be rewritten as a compound inequality: −a < x < a. Similarly, if the inequality is |x| ≤ a, it translates to −a ≤ x ≤ a. This approach allows the absolute value inequality to be expressed without absolute value symbols, making it easier to solve.

Before rewriting, it is essential to isolate the absolute value expression on one side of the inequality. For example, in an inequality like |x + 1| + 3 ≤ 5, subtracting 3 from both sides isolates the absolute value, resulting in |x + 1| ≤ 2. Then, the inequality can be rewritten as −2 ≤ x + 1 ≤ 2. Solving this three-part inequality involves performing the same operation on all three parts to isolate x. Subtracting 1 throughout yields −3 ≤ x ≤ 1, which can be expressed in interval notation as [−3, 1]. Graphically, this solution is represented by a line segment between −3 and 1, including both endpoints, often shown with closed circles or brackets to indicate inclusion.

It is important to recognize special cases based on the value of a in inequalities of the form |x| < a or |x| ≤ a. Since absolute value represents distance and cannot be negative, if a is negative, the inequality has no solution because an absolute value cannot be less than a negative number. If a = 0, the inequality |x| ≤ 0 implies that the absolute value must be exactly zero, so x = 0. Applying these rules helps avoid incorrect solutions and clarifies the nature of absolute value inequalities.

In summary, solving absolute value inequalities with less than or less than or equal to signs involves isolating the absolute value, rewriting the inequality as a compound inequality without absolute values, and then solving for the variable. Special attention must be given to cases where the boundary value is zero or negative, as these determine whether solutions exist or are unique. Mastery of these concepts enables effective problem-solving and a deeper understanding of how absolute value relates to distance and inequality constraints.

Relative error is a key concept used to measure the accuracy of a value compared to its expected or true value. It is often expressed as an absolute value inequality to capture the allowable deviation. For example, if a bag of chips is supposed to contain 20 grams, but the actual weight can vary with a relative error no greater than 0.2, we can use the relative error formula to determine the acceptable weight range.

The formula for relative error is given by the absolute value of the difference between the actual value and the expected value, divided by the expected value, which must be less than or equal to the maximum relative error:

Here, c represents the actual weight of the bag, c_t is the target or expected weight (20 grams), and m is the maximum relative error (0.2). Substituting these values, the inequality becomes:

This absolute value inequality can be rewritten as a compound inequality:

To simplify, convert decimals to fractions: \$0.2 = \frac{2}{10}$. Multiplying all parts of the inequality by the least common denominator, 20, clears the fractions:

Simplifying each term yields:

Adding 20 to all sides isolates c:

This means the actual weight of the bag of chips can range from 16 grams to 24 grams while maintaining a relative error within 0.2. Understanding how to manipulate absolute value inequalities and apply the relative error formula is essential for solving real-world problems involving measurement tolerances and accuracy.

Solving absolute value inequalities where the absolute value expression is greater than or greater than or equal to a number involves rewriting the inequality into two separate inequalities without absolute values. This approach builds on the skills used for solving absolute value equations and inequalities.

For an inequality of the form |x| > a, the absolute value represents the distance between x and zero. Saying that this distance is greater than a means that x must be either greater than a or less than -a. Thus, the inequality |x| > a can be rewritten as two inequalities:

\[x > a \quad \text{or} \quad x < -a\]

This rule also applies when the inequality includes equality, such as |x| ≥ a, which becomes:

\[x \geq a \quad \text{or} \quad x \leq -a\]

Before applying this rule, it is essential to isolate the absolute value expression on one side of the inequality. For example, if the inequality is |x + 1| + 3 ≥ 5, subtract 3 from both sides to isolate the absolute value:

\[|x + 1| \geq 2\]

Then rewrite as two inequalities:

\[x + 1 \geq 2 \quad \text{or} \quad x + 1 \leq -2\]

Solving each inequality separately:

\[x \geq 1 \quad \text{or} \quad x \leq -3\]

The solution set can be expressed in interval notation as:

\[(-\infty, -3] \cup [1, \infty)\]

Graphically, this corresponds to shading all values less than or equal to -3 and all values greater than or equal to 1, using closed circles or brackets to indicate inclusion of the boundary points.

It is important to consider special cases when the number a is zero or negative. Since absolute value represents distance and is always nonnegative, any inequality of the form |x| ≥ 0 or |x| ≥ -a (where a is negative) is always true for all real numbers. Therefore, the solution set in these cases is all real numbers.

In summary, solving absolute value inequalities with greater than or greater than or equal to symbols involves isolating the absolute value, rewriting the inequality as two separate inequalities, and solving each one. Special attention should be given to cases where the number compared to the absolute value is zero or negative, as these yield solutions that include all real numbers.

To solve the inequality involving the absolute value expression \(|3x + 6| > 0\), we apply the general rule for absolute value inequalities. Since the inequality is strictly greater than zero, it splits into two separate inequalities without the absolute value: one where the expression inside is greater than zero, and another where it is less than zero. Specifically, we write:

Solving the first inequality, subtract 6 from both sides:

then divide both sides by 3:

For the second inequality, similarly subtract 6:

and divide by 3:

Combining these results, the solution is all real numbers except \(x = -2\), since \(x\) must be either greater than or less than \(-2\) but not equal to it.

Expressing this solution in interval notation, we write the union of two intervals that exclude \(-2\):

Graphically, this is represented by two rays on the number line: one extending leftward from \(-2\) toward negative infinity, and the other extending rightward from \(-2\) toward positive infinity. At \(x = -2\), an open parenthesis or circle indicates that this point is not included in the solution set.

This approach highlights how absolute value inequalities can be transformed into compound inequalities, allowing for straightforward algebraic manipulation and clear graphical interpretation. Understanding this method is essential for solving a wide range of absolute value problems and expressing their solutions accurately in interval notation.