Question

Solve: \(4|x+1|-9<23\)

Short answer

-9 < x < 7

Step-by-step solution

- Isolate the Absolute Value Expression

Start by isolating the absolute value expression on one side of the inequality. Add 9 to both sides: \[ 4|x+1| - 9 + 9 < 23 + 9 \] This simplifies to: \[ 4|x+1| < 32 \]

- Divide Both Sides by 4

Next, divide both sides of the inequality by 4 to solve for the absolute value: \[ \frac{4|x+1|}{4} < \frac{32}{4} \] This simplifies to: \[ |x+1| < 8 \]

- Set Up Two Inequalities

Recall that if \( |A| < B \), then \( -B < A < B \). Therefore, set up the two inequalities: \[ -8 < x+1 < 8 \]

- Solve the Compound Inequality

Isolate \( x \) in the compound inequality by subtracting 1 from all parts: \[ -8 - 1 < x + 1 - 1 < 8 - 1 \] Simplifying each part: \[ -9 < x < 7 \]

Key concepts

Absolute Value

Absolute value is a concept that measures the distance of a number from zero on the number line, regardless of direction. It's always non-negative. For any number \(x\), its absolute value is written as \(|x|\). Some key points to remember about absolute value are:

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)

In the context of inequalities, absolute values can describe a range of values. For example, if \(|x| < 8\), it means that the distance of \(x\) from zero is less than 8. This translates to two separate inequalities: \(-8 < x < 8\), creating a compound inequality that we need to solve.

Inequalities

Inequalities are mathematical expressions involving the symbols \(<, \leq, >, \geq\). They indicate that one quantity is less than, less than or equal to, greater than, or greater than or equal to another quantity. Understanding inequalities is crucial when solving equations, especially those involving absolute values. For instance, the inequality \(4|x+1| - 9 < 23\) represents a range of outputs that 4 times the absolute value of \(x+1\), minus 9, can achieve and still satisfy the inequality. To solve it, you'll often need to manipulate the inequality while keeping the relation intact, like isolating the variable, as seen in the solution steps of the example problem.

Compound Inequalities

Compound inequalities involve two separate inequalities that are combined into one statement by the word 'and' or 'or'. When dealing with absolute values in inequalities, you often encounter compound inequalities. For example, if \(|A| < B\), it can be rewritten as \(-B < A < B\). This indicates that \(A\) is between \(-B\) and \(B\). When you solve for the variable in a compound inequality, you must consider both parts simultaneously. In the example problem, once we isolated the absolute value, we derived \(|x+1| < 8\). This led us to the compound inequality \(-8 < x+1 < 8\), which was further simplified to \(-9 < x < 7\) by isolating \(x\).

Isolation of Variables

Isolation of variables is a fundamental technique in solving equations and inequalities. It involves manipulating the equation or inequality to get the variable alone on one side. This makes it easier to see the value or range of values that the variable can take. In the given problem, we start by isolating the absolute value expression. We do this by performing operations like adding, subtracting, multiplying, or dividing both sides of the inequality by the same number, ensuring that the inequality remains balanced. Specifically, after isolating \(4|x+1|\) in the inequality \(4|x+1| < 32\), we divided both sides by 4 to simplify. Once the absolute value was isolated as \(|x+1| < 8\), we transformed it into the compound inequality \(-8 < x+1 < 8\) and then further isolated \(x\) by subtracting 1 from all parts, leading to the final solution range \(-9 < x < 7\).