An equation is a mathematical statement that two expressions are equivalent. The solution set of an equation is the value or values of the variable that make the equation true. A linear equation in one variable is an equation written in the form ax = b, where a and b are constants and a ≠ 0.

Linear Equations in One variable Nonlinear Equations 4x = 8 + 1 = 32 3x – = –9 + 1 = 41 2x – 5 = 0.1x +2 3 – 2x = –5 Notice that the variable in a linear equation is not under a radical sign and is not raised to a power other than 1. The variable is also not an exponent and is not in a denominator. Solving a linear equation requires isolating the variable on one side of the equation by using the properties of equality.

Properties of Equality Addition / Subtraction Property of Equality If you add or subtract the same quantity to or from both sides of an equation, the equation will still be true. Example:

Properties of Equality Multiplication / Division Property of Equality If you multiply or divide both sides of an equation by the same quantity, the equation will still be true. Example:

Solving Linear Equations If there are variables on both sides of the equation: simplify each side. (2) collect all variable terms on one side and all constants terms on the other side. (3) isolate the variables.

Example Method 1 Distribute before solving. 4(m + 12) = –36 4m + 48 = –36 Distribute 4. –48 –48 Subtract 48 from both sides. 4m = –84 = 4m –84 4 4 Divide both sides by 4. m = –21

Example Solve 4(m + 12) = –36 Method 2 The quantity (m + 12) is multiplied by 4, so divide by 4 first. 4(m + 12) = –36 Divide both sides by 4. 4 4 m + 12 = –9 –12 –12 Subtract 12 from both sides. m = –21

Example 1 Method 1 Distribute before solving. –3(5 – 4r) = –9. –15 + 12r = –9 Distribute 3. +15 +15 Add 15 to both sides. 12r = 6 = 12r 6 12 12 Divide both sides by 12. r =

Example 1 Solve –3(5 – 4r) = –9. Method 2 –3(5 – 4r) –9 –3 –3 = Divide both sides by –3. 5 – 4r = 3 –5 –5 Subtract 5 from both sides. –4r = –2 = –4 –4 –4r –2 Divide both sides by –4. r =

Example 2 Method 1 Distribute before solving. 3(2 – 3p) = 42 6 – 9p = 42 Distribute 3. –6 –6 Subtract 6 from both sides. –9p = 36 = –9p 36 –9 –9 Divide both sides by –9. p = –4

Example 2 Solve 3(2 –3p) = 42. Method 2 The quantity (2 – 3p) is multiplied by 3, so divide by 3 first. 3(2 – 3p) = 42 Divide both sides by 3. 3 3 2 – 3p = 14 –2 –2 Subtract 2 from both sides. –3p = 12 Divide both sides by –3. –3 –3 p = –4

Example 3: Solving Equations with Variables on Both Sides Solve 3k– 14k + 25 = 2 – 6k – 12. Simplify each side by combining like terms. –11k + 25 = –6k – 10 +11k +11k Collect variables on the right side. 25 = 5k – 10 Add. +10 + 10 Collect constants on the left side. 35 = 5k Isolate the variable. 5 5 7 = k

A contradiction is an equation that has no solutions. Example: 3v – 9 – 4v = –(5 + v). 3v – 9 – 4v = –(5 + v) –9 – v = –5 – v Simplify. + v + v –9 ≠ –5 x Contradiction The equation has no solution. The solution set is the empty set, which is represented by the symbol .

An identity is an equation that is true for all values of the variable. Example: 2(x – 6) = –5x – 12 + 7x. 2(x – 6) = –5x – 12 + 7x Simplify. 2x – 12 = 2x – 12 –2x –2x –12 = –12  Identity The solutions set is all real number, or .

An inequality is a statement that compares two expressions by using the symbols <, >, ≤, ≥, or ≠. The graph of an inequality is the solution set, the set of all points on the number line that satisfy the inequality. The properties of equality are true for inequalities, with one important difference. IMPORTANT: If you multiply or divide both sides by a NEGATIVE number, you must reverse the direction of inequality symbol. Example: -6x ≥ 30 -6 -6 x ≤ -5

These properties also apply to inequalities expressed with >, ≥, and ≤.

Example 5: Solving Inequalities Solve and graph 8a –2 ≥ 13a + 8. 8a – 2 ≥ 13a + 8 –13a –13a Subtract 13a from both sides. –5a – 2 ≥ 8 Add 2 to both sides. +2 +2 –5a ≥ 10 Divide both sides by –5 and reverse the inequality. –5a ≤ 10 –5 –5 a ≤ –2

  Example 5 Continued Solve and graph 8a – 2 ≥ 13a + 8. Check Test values in the original inequality. • –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 Test x = –4 Test x = –2 Test x = –1 8(–4) – 2 ≥ 13(–4) + 8 8(–2) – 2 ≥ 13(–2) + 8 8(–1) – 2 ≥ 13(–1) + 8 –34 ≥ –44  –18 ≥ –18  –10 ≥ –5 x So –4 is a solution. So –2 is a solution. So –1 is not a solution.

Check It Out! Example 5 Solve and graph x + 8 ≥ 4x + 17. x + 8 ≥ 4x + 17 –x –x Subtract x from both sides. 8 ≥ 3x +17 Subtract 17 from both sides. –17 –17 –9 ≥ 3x –9 ≥ 3x Divide both sides by 3. 3 3 –3 ≥ x or x ≤ –3

Check It Out! Example 5 Continued Solve and graph x + 8 ≥ 4x + 17. Check Test values in the original inequality. • –6 –5 –4 –3 –2 –1 0 1 2 3 Test x = –6 Test x = –3 Test x = 0 –6 + 8 ≥ 4(–6) + 17 –3 +8 ≥ 4(–3) + 17 0 +8 ≥ 4(0) + 17 2 ≥ –7  5 ≥ 5  8 ≥ 17 x So –6 is a solution. So –3 is a solution. So 0 is not a solution.