Lesson 2.7: Solve Absolute Value Inequalities

New Concept

Mastering absolute value means understanding distance from zero. In this lesson, we'll apply this idea to solve equations and inequalities, finding values that are a specific distance, within a range, or beyond a certain point from zero.

What’s next

You'll work through interactive examples of solving absolute value equations and inequalities, and then apply your skills to real-world problems on our platform.

Property

The absolute value of a number is its distance from zero on the number line. The absolute value of a number $n$ is written as $|n|$ and $|n| \geq 0$ for all numbers. Absolute values are always greater than or equal to zero.

Examples

• The absolute value of $-9$ is $9$, because $-9$ is $9$ units away from $0$. This is written as $|-9| = 9$.
• The absolute value of $25$ is $25$, because $25$ is $25$ units away from $0$. This is written as $|25| = 25$.
• The equation $|x| = -4$ has no solution. Absolute value represents distance, which cannot be a negative number.

Explanation

Think of absolute value as a 'distance-meter' from zero. Since distance can't be negative, the absolute value of any number, positive or negative, will always be a positive result or zero. It simply tells you how far away you are.

Property

For any algebraic expression, $u$, and any positive real number, $a$, if $|u| = a$, then $u = -a$ or $u = a$. To solve an absolute value equation, first isolate the absolute value expression. Then, write two equivalent equations and solve each one separately.

Examples

• To solve $|3x - 1| - 5 = 10$, first isolate the absolute value: $|3x - 1| = 15$. Then set up two equations: $3x - 1 = 15$ or $3x - 1 = -15$. Solving gives $x = \frac{16}{3}$ or $x = -\frac{14}{3}$.
• The equation $3|x - 8| + 11 = 5$ simplifies to $3|x - 8| = -6$, or $|x - 8| = -2$. Since an absolute value cannot be negative, there is no solution.
• To solve $|x + 4| = |2x - 2|$, set up two cases: $x + 4 = 2x - 2$ or $x + 4 = -(2x - 2)$. Solving these yields $x = 6$ or $x = -\frac{2}{3}$.

Explanation

If the absolute value of something is a positive number $a$, it means the 'something' inside is either $a$ units to the right of zero or $a$ units to the left. This is why you must solve two separate cases.

Property

To solve an absolute value inequality with a 'less than' or 'less than or equal to' sign, we rewrite it as a compound inequality. If $|u| < a$, the equivalent inequality is $-a < u < a$. This also applies for $\leq$. First, isolate the absolute value expression, then write the equivalent compound inequality and solve.

Examples

• To solve $|x| < 9$, you are looking for all numbers whose distance from zero is less than 9. The solution is $-9 < x < 9$, which is $(-9, 9)$ in interval notation.
• To solve $|2x - 5| \leq 3$, rewrite it as $-3 \leq 2x - 5 \leq 3$. Adding 5 to all parts gives $2 \leq 2x \leq 8$. Dividing by 2 gives $1 \leq x \leq 4$, or $[1, 4]$.
• The inequality $|x + 4| + 5 < 2$ simplifies to $|x + 4| < -3$. Since an absolute value can never be less than a negative number, there is no solution.

Explanation

When an absolute value is less than a number, it means the expression inside is 'close' to zero. Its distance from zero is small, so its value must be 'sandwiched' between the negative and positive boundaries of that distance.

Property

To solve an absolute value inequality with a 'greater than' or 'greater than or equal to' sign, we rewrite it as two separate inequalities. If $|u| > a$, the equivalent inequality is $u < -a$ or $u > a$. This also applies for $\geq$. First, isolate the absolute value, then write and solve the two separate inequalities.

Examples

• To solve $|x| > 5$, you are looking for numbers whose distance from zero is more than 5. The solution is $x < -5$ or $x > 5$. In interval notation, this is $(-\infty, -5) \cup (5, \infty)$.
• To solve $|2x - 1| \geq 7$, write two inequalities: $2x - 1 \leq -7$ or $2x - 1 \geq 7$. Solving these gives $x \leq -3$ or $x \geq 4$. The interval is $(-\infty, -3] \cup [4, \infty)$.
• The inequality $|x - 9| > -2$ is true for all real numbers. Since absolute value is always zero or positive, it is always greater than any negative number. The solution is $(-\infty, \infty)$.

Explanation

When an absolute value is greater than a number, it means the expression inside is 'far' from zero. Its value must be in one of two regions: either far to the left (very negative) or far to the right (very positive).

Property

Absolute value inequalities can model situations involving a tolerance or an acceptable range of variation from an ideal value. The general form is $|\text{actual} - \text{ideal}| \leq \text{tolerance}$. Let $x$ be the actual measurement, $i$ be the ideal measurement, and $t$ be the tolerance. The inequality is $|x - i| \leq t$.

Examples

• A machine part should be 50 mm long, with a tolerance of 0.02 mm. Let $x$ be the actual length. The situation is modeled by $|x - 50| \leq 0.02$. This gives the acceptable range $49.98 \leq x \leq 50.02$ mm.
• The ideal temperature for a greenhouse is 75 degrees Fahrenheit. The actual temperature can vary by up to 3 degrees. The acceptable range is found by solving $|T - 75| \leq 3$, which gives $72 \leq T \leq 78$ degrees.
• A bag of chips should weigh 16 ounces, with an allowed variance of 0.25 ounces. The acceptable weight range can be found with $|w - 16| \leq 0.25$. The solution is $15.75 \leq w \leq 16.25$ ounces.

Explanation

Use absolute value to describe a margin of error. It measures the difference between what you have (actual) and what you want (ideal), ensuring that this difference, regardless of direction, stays within an acceptable limit (tolerance).