Problem

Solve the equation, and check your solution. See Examples.EXAMPLE Applying Both Properties...

Solve the equation, and check your solution. See Examples.

EXAMPLE Applying Both Properties of Equality to Solve an Equation
Solve − 6x + 5 = 17.
Step 1 There are no parentheses, fractions, or decimals in this equation, so this step is not necessary.
Step 2
Step 3
Step 4 Check by substituting − 2 for x in the original equation.
The solution, − 2, checks, so the solution set is {−2}. now try
EXAMPLE Applying Both Properties of Equality to Solve an Equation
Solve 3x + 2 = 5x − 8.
Step 1 There are no parentheses, fractions, or decimals in the equation.
Step 2
Step 3
Step 4 Check by substituting 5 for x in the original equation.
The solution, 5, checks, so the solution set is {5}.
EXAMPLE Using the Four Steps to Solve an Equation
Solve 4(k − 3) − k = k − 6.
Step 1 Clear parentheses using the distributive property.
Step 2
Step 3
Step 4
The solution set of the equation is {3}.
EXAMPLE Using the Four Steps to Solve an Equation
Solve 8z − (3 + 2z) = 3z + 1.
Step 1
Step 2
Step 3
Step 4 Check that is the solution set.
EXAMPLE Using the Four Steps to Solve an Equation
Solve 4(4 −3x)= 32 − 8(x + 2).
Step 1
Step 2
Step 3
Step 4
Since the solution 0 checks, the solution set is {0}.
EXAMPLE Solving an Equation That Has Infinitely Many Solutions
Solve 5x − 15 = 5(x − 3).
Since the last statement (0 = 0) is true, any real number is a solution. We could have predicted this from the second line in the solution,
Try several values for x in the original equation to see that they all satisfy it.
An equation with both sides exactly the same, like 0 = 0, is called an identity. An identity is true for all replacements of the variables. As shown above, we write the solution set as {all real numbers}.
EXAMPLE Solving an Equation That Has No Solution
Solve 2x + 3(x + 1) = 5x + 4.
A false statement (3 = 4) results. The original equation, called a contradiction, has no solution. Its solution set is the empty set, or null set, symbolized.
EXAMPLE Solving an Equation with Decimals as Coefficients Solve 0.1/ + 0.05 (20 − t)= 0.09 (20).
Step 1 The decimals here are expressed as tenths (0.1) and hundredths (0.05 and 0.09). We choose the least exponent on 10 needed to eliminate the decimals. Here, we use 102 = 100.
Step 2
Step 3
Step 4 Check to confirm that {16} is the solution set.
EXAMPLE Solving an Equation That Has Infinitely Many Solutions
Solve 5x − 15 = 5(x − 3).
Since the last statement (0 = 0) is true, any real number is a solution. We could have predicted this from the second line in the solution,
Try several values for x in the original equation to see that they all satisfy it.
An equation with both sides exactly the same, like 0 = 0, is called an identity. An identity is true for all replacements of the variables. As shown above, we write the solution set as {all real numbers}.
EXAMPLE Solving an Equation That Has No Solution
Solve 2x + 3(x + 1) = 5x + 4.
A false statement (3 = 4) results. The original equation, called a contradiction, has no solution. Its solution set is the empty set, or null set, symbolized.