16. Solve the equation and inequality for x.
Part A: Solve the equation for x showing all steps for credit: ax−5=18
Part B: Solve the inequality for x showing all steps for credit: ax−5>18
Part C: State the difference between an equation and an inequality regarding solving steps and solutions.
(6 points)
Part A:
To solve the equation ax - 5 = 18, we need to isolate the variable x by moving all other terms to the other side of the equation.
1. Start with ax - 5 = 18
2. Add 5 to both sides of the equation: ax - 5 + 5 = 18 + 5
3. Simplify: ax = 23
4. Divide both sides of the equation by a: (ax)/a = 23/a
5. Simplify: x = 23/a
The solution for x is x = 23/a.
Part B:
To solve the inequality ax - 5 > 18, we need to isolate the variable x, while keeping in mind that the inequality symbol will change direction if we multiply or divide by a negative number.
1. Start with ax - 5 > 18
2. Add 5 to both sides of the inequality: ax - 5 + 5 > 18 + 5
3. Simplify: ax > 23
4. Divide both sides of the inequality by a, paying attention to whether a is positive or negative. If a is positive: (ax)/a > 23/a. If a is negative: (ax)/a < 23/a.
5. Simplify: x > 23/a (if a is positive) or x < 23/a (if a is negative)
The solution for x depends on the value of a:
- If a is positive, the solution is x > 23/a.
- If a is negative, the solution is x < 23/a.
Part C:
The main difference between an equation and an inequality is how they are solved and the type of solutions they yield.
For an equation, the goal is to find a value or set of values that make the equation true. The solving steps for an equation involve isolating the variable on one side of the equation by performing the same operation on both sides. The solution for an equation is usually a specific value or a set of values that satisfy the equation.
For an inequality, the goal is to find a range of values that make the inequality true. The solving steps for an inequality involve isolating the variable on one side of the inequality by performing the same operation on both sides. However, when multiplying or dividing by a negative number, the inequality symbol needs to be flipped. The solution for an inequality is a range of values that satisfy the inequality, usually expressed using inequality notation.
Part A:
To solve the equation ax - 5 = 18, we need to isolate the variable x. Here are the steps:
Step 1: Add 5 to both sides of the equation:
ax - 5 + 5 = 18 + 5
ax = 23
Step 2: Divide both sides of the equation by a:
ax/a = 23/a
x = 23/a
Therefore, the solution to the equation ax - 5 = 18 is x = 23/a.
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Part B:
To solve the inequality ax - 5 > 18, we need to isolate the variable x. Here are the steps:
Step 1: Add 5 to both sides of the inequality:
ax - 5 + 5 > 18 + 5
ax > 23
Step 2: Divide both sides of the inequality by a. However, if a is negative, the inequality sign needs to be flipped:
If a > 0:
ax/a > 23/a
x > 23/a
If a < 0:
ax/a < 23/a
x < 23/a
Therefore, the solution to the inequality ax - 5 > 18 is x > 23/a if a > 0, or x < 23/a if a < 0.
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Part C:
The difference between an equation and an inequality lies in the solutions and the approach to solving them.
In an equation, the goal is to find the specific value or values of the variable that make the equation true. Solving an equation involves performing arithmetic operations to isolate the variable on one side of the equation.
In an inequality, the goal is to find the range of values of the variable that make the inequality true. Solving an inequality involves performing arithmetic operations to isolate the variable on one side of the inequality, but we also need to pay attention to the sign of the inequality. If at any point we multiply or divide the inequality by a negative number, the inequality sign needs to be flipped.
Therefore, the difference between an equation and an inequality lies in the range of possible solutions and the additional consideration for signs in an inequality.