## Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

### Chapter 14. Solving Linear Equations and Inequalities

A linear equation in one variable, say *x*, has the standard form *ax* + *b* = *c*, *a* ≠ 0, where *a*, *b*, and *c* are constants. For example, 3*x* – 7 = 14 is a linear equation in standard form. An equation has two sides. The expression on the left side of the equal sign is the *left side* of the equation, and the expression on the right side of the equal sign is the *right side* of the equation.

**Solving One-Variable Linear Equations**

To *solve a linear equation* that has one variable *x* means to find a numerical value for *x* that makes the equation true. An equation is true when the left side has the same value as the right side. When you solve an equation, you undo what has been done to *x* until you get an expression like this: *x* = a number. As you proceed, you exploit the fact that addition and subtraction undo each other; and, similarly, multiplication and division undo one another.

The goal in solving a linear equation is to get the variable by itself on only one side of the equation and with a coefficient of 1 (usually understood).

You solve an equation using the properties of real numbers and simple algebraic tools. An equation is like a balance scale. To keep the equation in balance, when you do something to one side of the equation, you must do to the same thing to the other side of the equation.

**Tools for Solving Linear Equations**

Add the same number to both sides. Subtract the same number from both sides. Multiply both sides by the same *nonzero* number.

Divide both sides by the same *nonzero* number.

It is important to remember that when you are solving an equation, you must *never* multiply or divide both sides by 0.

**Problem** Solve the equation.

**e**.

**Solution**

*Step 1*. The variable appears on both sides of the equation, so subtract 3*x* from the right side to remove it from that side. To maintain balance, subtract 3*x* from the left side, too.

*Step 2*. Simplify both sides by combining like variable terms.

*Step 3*. 9 is added to the variable term, so subtract 9 from both sides.

*Step 4*. Simplify both sides by combining constant terms.

*Step 5*. You want the coefficient of *x* to be 1, so divide both sides by 2.

*Step 6*. Simplify.

*Step 7*. Check your answer by substituting –5 for *x* in the original equation,

Substitute –5 for *x* on the left side of the equation: Similarly, on the right side, you have Both sides equal –16, so –5 is the correct answer.

*Step 1*. Use the distributive property to remove parentheses.

*Step 2*. 24 is subtracted from the variable term, so add 24 to both sides.

*Step 3*. Simplify both sides by combining constant terms.

**4 x = 64**

*Step 4*. You want the coefficient of *x* to be 1, so divide both sides by 4.

*Step 5*. Simplify.

*x* = 16

*Step 6*. Check your answer by substituting 16 for *x* in the original equation, 4(*x* – 6) = 40.

Substitute 16 for *x* on the left side of the equation: On the right side, you have 40 as well. Both sides equal 40, so 16 is the correct answer.

*Step 1*. 7 is subtracted from the variable term, so add 7 to both sides.

*Step 2*. Simplify both sides by combining constant terms.

–**3 x = 21**

*Step 3*. You want the coefficient of *x* to be 1, so divide both sides by –3.

*Step 4*. Simplify.

*x* = –7

*Step 5*. Check your answer by substituting –7 for *x* in the original equation, – 3*x* – 7 = 14.

Substitute –7 for *x* on the left side of the equation: –3*x* – 7 = –3(–7) – 7 = 21 – 7 = 14. On the right side, you have 14 as well. Both sides equal 14, so –7 is the correct answer.

*Step 1*. The variable appears on both sides of the equation, so add 2*x* to the right side to remove it from that side. To maintain balance, add 2*x* to the left side, too.

*Step 2*. Simplify both sides by combining like variable terms.

*Step 3*. 2 is subtracted from the variable term, so add 2 to both sides.

*Step 4*. Simplify both sides by combining constant terms.

**5 x = 9**

*Step 5*. You want the coefficient of *x* to be 1, so divide both sides by 5.

*Step 6*. Simplify.

*x* = 1.8

*Step 7*. Check your answer by substituting 1.8 for *x* in the original equation,

Substitute 1.8 for *x* on the left side of the equation: Similarly, on the right side, you have Both sides equal 3.4, so 1.8 is the correct answer.

**e**.

*Step 1*. Eliminate fractions by multiplying both sides by 10, the least common multiple of 2 and 5. Write 10 as to avoid errors.

*Step 2*. Simplify.

**5( x – 3) = 2(2x + 4)**

**5 x – 15 = 4x + 8**

*Step 3*. The variable appears on both sides of the equation, so subtract 4*x* from the right side to remove it from that side. To maintain balance, subtract 4*x* from the left side, too.

*Step 4*. Simplify both sides by combining variable terms.

*Step 5*. 15 is subtracted from the variable term, so add 15 to both sides.

*Step 6*. Simplify both sides by combining constant terms.

*x* = 23

*Step 7*. Check your answer by substituting 23 for *x* in the original equation, .

Substitute 23 for *x* on the left side of the equation:

. Similarly, on the right side, you have .

. Both sides equal 10, so 23 is the correct answer.

**Solving Two-Variable Linear Equations for a Specific Variable**

You can use the procedures for solving a linear equation in one variable *x* to solve a two-variable linear equation, such as 6*x* + 2*y* = 10, for one of the variables in terms of the other variable. As you solve for the variable of interest, you simply treat the other variable as you would a constant. Often, you need to solve for *y* to facilitate the graphing of an equation. (See __Chapter 17__ for a fuller discussion of this topic.) Here is an example.

**Problem** Solve 6*x* + 2*y* = 10 for *y*.

**Solution**

*Step 1*. 6*x* is added to the variable term 2*y*, so subtract 6*x* from both sides.

When you are solving 6*x* 2*y* = 10 for *y*, treat 6*x* as if it were a constant.

*Step 2*. Simplify.

*Step 3*. You want the coefficient of *y* to be 1, so divide both sides by 2.

You must divide *both* terms of the numerator by 2.

*Step 4*. Simplify.

*y* = 5 – 3*x*

**Solving Linear Inequalities**

If you replace the equal sign in a linear equation with <, >, ≤, or ≥, the result is a linear inequality. You solve linear inequalities just about the same way you solve equations. There is just one important difference. When you multiply or divide both sides of an inequality by a *negative* number, you must *reverse* the direction of the inequality symbol. To help you understand why you must do this, consider the two numbers, 8 and 2. You know that 8 > 2 is a true inequality because 8 is to the right of 2 on the number line, as shown in __Figure 14.1__.

**Figure 14.1** The numbers 2 and 8 on the number line

If you multiply both sides of the inequality 8 > 2 by a negative number, say, –1, you must reverse the direction of the inequality so that you will still have a true inequality, namely, –8 < –2. You can verify that –8 < –2 is a true inequality by observing that –2 is to the right of –8 on the number line as shown in __Figure 14.2__.

**Figure 14.2** The numbers –8 and –2 on the number line

If you neglect to reverse the direction of the inequality symbol after multiplying both sides of 8 > 2 by –1, you get the false inequality –8 > –2.

**Problem** Solve the inequality.

**Solution**

*Step 1*. The variable appears on both sides of the inequality, so subtract 3*x* from the right side to remove it from that side. To maintain balance, subtract 3*x* from the left side, too.

*Step 2*. Simplify both sides by combining like variable terms.

*Step 3*. 6 is added to the variable term, so subtract 6 from both sides.

When solving an inequality, do *not* reverse the direction of the inequality symbol because of subtracting the same number from both sides.

*Step 4*. Simplify both sides by combining constant terms.

*Step 5*. You want the coefficient of *x* to be 1, so divide both sides by 2.

When solving an inequality, do *not* reverse the direction of the inequality symbol because of dividing both sides by a positive number.

*Step 6*. Simplify.

** x < –4** is the answer.

*Step 1*. Use the distributive property to remove parentheses.

*Step 2*. 24 is subtracted from the variable term, so add 24 to both sides.

When solving an inequality, do *not* reverse the direction of the inequality because of adding the same number to both sides.

*Step 3*. Simplify both sides by combining constant terms.

*Step 4*. You want the coefficient of *x* to be 1, so divide both sides by 4.

*Step 5*. Simplify.

** x ≥ 17** is the answer.

*Step 1*. 7 is subtracted from the variable term, so add 7 to both sides.

*Step 2*. Simplify both sides by combining constant terms.

–*3x* > 21

*Step 3*. You want the coefficient of *x* to be 1, so divide both sides by –3 and *reverse the direction of the inequality because you divided by a negative number.*

When solving an inequality, remember to reverse the direction of the inequality when you divide both sides by the same negative number.

*Step 4*. Simplify.

** x < –7** is the answer.

**Exercise 14**

*For 1–5, solve the equation for x*.

__4__.

*For 7–10, solve the inequality for x*.

__10__.