When an absolute value equation contains extra terms, apply algebraic properties to completely isolate the absolute value expression before splitting it into two equations. For example, to solve $$2|x - 7| + 5 = 9$$, first subtract $$5$$ from both sides to obtain $$2|x - 7| = 4$$. Next, divide by $$2$$ to yield the isolated equation $$|x - 7| = 2$$. Then, formulate the equivalent equations: $$x - 7 = -2$$ or $$x - 7 = 2$$. Solving each produces $$x = 5$$ or $$x = 9$$. Finally, substituting both values back into the initial equation verifies that both $$x = 5$$ and $$x = 9$$ are valid solutions.

Example: Solving Absolute Value Equations Requiring Isolation

If isolating the absolute value expression results in an equation where the absolute value equals a negative number, the equation possesses no solution. For example, to solve $$|\frac{2}{3}x - 4| + 11 = 3$$, isolating the absolute value by subtracting $$11$$ from both sides yields $$|\frac{2}{3}x - 4| = -8$$. Because the distance represented by an absolute value can never be negative, this statement is mathematically false for any value of $$x$$. Thus, finding it equal to a negative constant immediately indicates that there is no solution.

Example: Absolute Value Equations with No Solution

To solve an absolute value equation, the absolute value expression must first be isolated. For example, to solve $$|5x - 4| - 3 = 8$$, add $$3$$ to both sides to isolate the absolute value, yielding $$|5x - 4| = 11$$. Next, write the equivalent equations: $$5x - 4 = -11$$ and $$5x - 4 = 11$$. Solving each equation by adding $$4$$ and dividing by $$5$$ gives the solutions $$x = -\frac{7}{5}$$ and $$x = 3$$. Both solutions should be checked by substituting them back into the original equation to verify they are valid.

Example: Isolating the Absolute Value Expression

To actively practice solving absolute value equations requiring prior isolation, solve the given equations $$|3x - 5| - 1 = 6$$ and $$|4x - 3| - 5 = 2$$. For the first equation, isolate the absolute value by adding $$1$$ to both sides, which results in $$|3x - 5| = 7$$. Generating the equivalent equations yields $$3x - 5 = -7$$ or $$3x - 5 = 7$$. Adding $$5$$ and dividing by $$3$$ for each case provides the solutions $$x = -\frac{2}{3}$$ or $$x = 4$$. For the second equation, adding $$5$$ to both sides isolates the absolute value, producing $$|4x - 3| = 7$$. The equivalent equations are $$4x - 3 = -7$$ or $$4x - 3 = 7$$. Adding $$3$$ and dividing by $$4$$ evaluates to the final solutions $$x = -1$$ or $$x = \frac{5}{2}$$.

Try It: Solving $$|3x - 5| - 1 = 6$$ and $$|4x - 3| - 5 = 2$$

To solve an absolute value equation, follow a structured four-step procedure. First, isolate the absolute value expression on one side of the equation using standard algebraic properties. Second, write the equivalent equations by setting the algebraic expression inside the absolute value equal to both the positive and negative versions of the isolated constant. Third, solve each resulting equation individually. Finally, check each solution by substituting it back into the original absolute value equation to verify that it results in a true mathematical statement.

Claude

**Solving an equation** is the purposeful process of discovering the specific value or values of a variable that make the equation a true mathematical statement. Conceptually analogous to solving a puzzle, the ultimate goal of this procedure is to identify the solution that accurately completes the logical structure of the equation.

Solving an Equation

For any algebraic expression $$u$$ and any positive real number $$a$$, the condition $$|u| = a$$ means that the mathematical statement can be rewritten as two equivalent equations: $$u = -a$$ or $$u = a$$. This property stems from the fact that we are seeking all values whose distance from zero is $$a$$, and both $$a$$ and $$-a$$ satisfy this requirement. A single statement for the solutions can also be written as $$u = \pm a$$. Remember that an absolute value cannot be a negative number.

Absolute Value Equations

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OpenStax Intermediate Algebra

To conceptually clarify the process of solving an algebraic equation, a physical model using envelopes and counters can be employed. In this model, an envelope represents the unknown variable because its exact contents are hidden, and each counter represents one unit (the number $$1$$). An equation is modeled by arranging envelopes and counters on both sides of a workspace that represents the two sides of the equal sign. Because the two sides of an equation must be equal, the total value on each side of the workspace must match, even though some counters are hidden inside the envelopes. For equations of the form $$x + a = b$$ or $$x - a = b$$, a single envelope is placed alongside counters, and solving involves adding or removing counters from both sides to isolate the envelope. The model also extends to equations where the variable is multiplied by a constant, such as $$2x = 6$$: multiple identical envelopes on one side represent the repeated variable, and dividing the counters on the other side into equal groups — one group per envelope — reveals how many counters each envelope contains.

Envelope and Counter Model for Equations

The **Subtraction Property of Equality** states that when the same quantity is subtracted from both sides of an equation, the two sides remain equal. Mathematically, for any numbers $$a$$, $$b$$, and $$c$$:

If $$a = b$$,
then $$a - c = b - c$$

This property is a fundamental mathematical tool for solving equations. By applying it, one can systematically remove unwanted terms from one side of an equation by subtracting them from both sides, which helps to isolate the unknown variable by itself.

Subtraction Property of Equality

The **Addition Property of Equality** states that if the same quantity is added to both sides of an equation, the two sides remain equal. Mathematically, for any numbers $$a$$, $$b$$, and $$c$$:

If $$a = b$$,
then $$a + c = b + c$$

This property is a fundamental tool used to solve equations where a number is subtracted from the variable. To isolate the variable and "undo" the subtraction, the same number is added to both sides. For example, when solving an equation like $$x - 5 = 8$$, adding $$5$$ to both sides isolates $$x$$ while maintaining the equality.

Addition Property of Equality

While some equations allow the variable to be isolated with just one operation, most algebraic equations require multiple steps to solve. Typically, this process requires simplifying one or both sides of the equation as much as possible before applying a property of equality — such as the Addition, Subtraction, Division, or Multiplication Property — to isolate the variable. The most common simplification techniques are **combining like terms** and **using the distributive property**. Simplification should be carried out for each side of the equation independently. It is important to recognize that simplification is a distinct process from solving: simplifying means reducing each side's expression to its simplest form, while solving means applying the same operation to both sides simultaneously to isolate the variable.

Simplifying Before Solving an Equation

To solve a linear equation, follow a systematic five-step, general strategy:

1. Simplify both sides: Simplify each side of the equation as much as possible individually. Use the Distributive Property to remove any parentheses, and combine like terms.
2. Isolate the variable terms: Collect all the variable terms on one side of the equation. This is achieved by using the Addition or Subtraction Property of Equality.
3. Isolate the constant terms: Collect all the constant terms on the other side of the equation. Again, use the Addition or Subtraction Property of Equality.
4. Solve for the variable: Make the coefficient of the variable term equal to $$1$$. Use the Multiplication or Division Property of Equality, then formally state the solution to the equation.
5. Check the answer: Verify the solution by substituting it back into the original equation to ensure the result simplifies to a true mathematical statement.

Strategy for Solving Linear Equations

The **Division Property of Equality** states that dividing both sides of an equation by the same nonzero number preserves the equality. For any numbers $$a$$, $$b$$, and $$c$$ with $$c \neq 0$$:

If $$a = b$$,
then $$\frac{a}{c} = \frac{b}{c}$$

This property is used to solve equations in which the variable is multiplied by a coefficient. To isolate the variable and "undo" the multiplication, both sides are divided by that coefficient. For example, when solving an equation like $$7x = 56$$, dividing both sides by $$7$$ yields $$x = 8$$, because $$\frac{7x}{7} = x$$ and $$\frac{56}{7} = 8$$. The restriction $$c \neq 0$$ is essential because division by zero is undefined.

Division Property of Equality

The **Multiplication Property of Equality** states that multiplying both sides of an equation by the same number preserves the equality. For any numbers $$a$$, $$b$$, and $$c$$:

If $$a = b$$,
then $$a \cdot c = b \cdot c$$

This property is used to solve equations in which the variable is divided by a number or has a fractional coefficient. Multiplying both sides by the appropriate value "undoes" the division or fraction. For example, to solve $$\frac{x}{3} = 5$$, multiply both sides by $$3$$ to obtain $$x = 15$$. Similarly, when the coefficient is a fraction such as $$\frac{2}{3}x = 10$$, multiplying both sides by the reciprocal $$\frac{3}{2}$$ isolates the variable: $$x = 10 \cdot \frac{3}{2} = 15$$. This technique leverages the inverse property of multiplication, since a number multiplied by its reciprocal equals $$1$$.

Multiplication Property of Equality

**Solving a formula for a specific variable** (sometimes called solving a **literal equation**) is the process of rearranging a formula so that one particular variable is isolated on one side of the equals sign with a coefficient of $$1$$, while all other variables and constants appear on the other side. The same properties of equality used to solve ordinary one-variable equations (Addition, Subtraction, Multiplication, and Division) are applied, but instead of producing a single numerical answer, the result expresses the chosen variable in terms of the remaining variables.

For instance, given the formula $$d = rt$$, solving for $$t$$ requires dividing both sides by $$r$$:

$$\frac{d}{r} = \frac{rt}{r}$$

$$t = \frac{d}{r}$$

The key strategy is to treat every letter other than the target variable as though it were a number, then use the standard equation-solving steps — simplify each side, collect terms, and isolate the variable — until the desired variable stands alone with a coefficient of $$1$$. This skill is essential whenever a formula must be evaluated repeatedly for different unknowns, because rearranging the formula once avoids repeating the algebraic manipulation each time a new value is needed.

Solving a Formula for a Specific Variable

To solve an equation that contains a variable under a square root, follow a four-step procedure:

**Step 1.** **Isolate the radical** on one side of the equation. If the square root expression is not already by itself, use addition, subtraction, or other algebraic operations to move all other terms to the opposite side.

**Step 2.** **Square both sides** of the equation. Applying the Squaring Property of Square Roots removes the radical from the isolated side, converting the equation into a standard polynomial equation.

**Step 3.** **Solve the new equation.** Use appropriate algebraic techniques (such as combining like terms, factoring, or applying properties of equality) to find the value(s) of the variable.

**Step 4.** **Check the answer.** Substitute each candidate solution back into the **original** radical equation and verify that it produces a true statement. Any value that does not satisfy the original equation is an extraneous solution and must be discarded.

How to Solve a Radical Equation

Solve the radical equation $$\sqrt{2x - 1} = 7$$ using the four-step procedure.

**Step 1 — Isolate the radical.** The square root $$\sqrt{2x - 1}$$ is already alone on the left side, so no rearrangement is needed.

**Step 2 — Square both sides.** Apply the Squaring Property to eliminate the radical:

$$(\sqrt{2x - 1})^2 = 7^2$$

$$2x - 1 = 49$$

**Step 3 — Solve the new equation.** Add $$1$$ to both sides: $$2x = 50$$. Divide both sides by $$2$$: $$x = 25$$.

**Step 4 — Check.** Substitute $$x = 25$$ into the original equation:

$$\sqrt{2(25) - 1} = \sqrt{50 - 1} = \sqrt{49} = 7$$

Since $$7 = 7$$ is true, $$x = 25$$ is confirmed as the solution.

Solving $$\sqrt{2x-1} = 7$$

When an office manager uses a formula to calculate a missing expense, the primary mathematical goal is to ____ the unknown variable by itself on one side of the equation.

In a professional context, such as determining the unit price needed to meet a specific revenue goal, which of the following best describes the process of 'solving' the resulting algebraic equation?

An administrative assistant is using an algebraic equation to find a missing value in a supply order. Match the following terms related to 'solving an equation' with their correct descriptions.

An administrative assistant is using an algebraic equation to find a missing value in a supply order. According to the definition of 'solving an equation,' what is the specific mathematical goal regarding the relationship between the two sides of the equation?

The Mathematical Objective of Solving Business Equations

In a workplace scenario, such as calculating the number of hours an employee worked based on their total earnings, the ultimate mathematical purpose of solving the resulting equation is to find a specific value for the variable that creates a true statement.

A project manager is solving an algebraic equation to determine the number of additional staff members needed for a new contract. Arrange the following conceptual stages of 'solving an equation' in the correct logical order, according to the standard mathematical definition.

Based on the formal definition of 'solving an equation,' state the primary mathematical goal the specialist has for the unknown variable (the hourly rate). Additionally, identify the specific condition that the final value must satisfy for the equation to be considered correctly solved.

Auditing Payroll Discrepancies

Imagine you are training a new administrative assistant on how to use algebraic formulas to find missing data in company records. Based on the mathematical definition provided in your training, write a brief response explaining what it means to 'solve' an equation. Your explanation must identify the primary mathematical goal concerning the variable and describe the ultimate purpose of the process in terms of the equation's sides.

The Purpose and Process of Solving Professional Equations

A laboratory technician is solving an equation to determine the exact amount of a catalyst needed for a chemical reaction. According to the definition of solving an equation, what is the defining characteristic of the specific value (the solution) the technician is searching for?

A project coordinator is reviewing the process of finding unknown values in a budget report. Match each aspect of the mathematical process of 'solving an equation' with its correct description according to the standard definition.

Procedure for Solving an Absolute Value Equation

For any algebraic expressions $$u$$ and $$v$$, if $$|u| = |v|$$, then $$u = -v$$ or $$u = v$$. This property demonstrates that when two absolute value expressions are equal to each other, the expressions inside the absolute value bars must logically either be exactly equal to each other or be exact opposites of each other.

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