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.
Part A:
To solve the equation ax - 5 = 18, we need to isolate the variable x by undoing the operations in reverse order.
First, we can add 5 to both sides of the equation:
ax - 5 + 5 = 18 + 5
ax = 23
Next, we divide both sides by a to solve for x:
(ax) / a = 23 / a
x = 23 / a
Therefore, the solution for x is x = 23 / a.
Part B:
To solve the inequality ax - 5 > 18, we again need to isolate the variable x.
First, we can add 5 to both sides of the equation:
ax - 5 + 5 > 18 + 5
ax > 23
Next, we divide both sides by a to solve for x:
(ax) / a > 23 / a
x > 23 / a
Therefore, the solution for the inequality is x > 23 / a.
Part C:
The difference between an equation and an inequality lies in the solutions.
An equation has a single value or set of values that make both sides of the equation equal. Once the equation is solved, the solution(s) will give a specific numerical value or values for the variable.
An inequality, on the other hand, represents a range of values that satisfy the inequality. Once the inequality is solved, the solution(s) will give a range of values or an interval for the variable that makes the inequality true.
Therefore, when solving an equation, we find specific values for the variable, while when solving an inequality, we find a range of values for the variable that satisfies the inequality.
Part A: Solve the equation ax - 5 = 18:
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
Part B: Solve the inequality ax - 5 > 18:
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 (assuming a is positive)
(ax) / a > 23 / a
x > 23/a
Part C: The difference between an equation and an inequality regarding solving steps and solutions is as follows:
- An equation is a statement of equality where we try to find the value(s) of the variable(s) that make both sides of the equation equal.
- To solve an equation, we perform the same operation on both sides, simplifying the equation until we isolate the variable.
- The solution to an equation is a single value or a set of values that satisfy the equation.
- An inequality, on the other hand, is a statement of inequality where we try to find the range of values that make the inequality true.
- When solving an inequality, we perform operations to isolate the variable, but we also need to consider the direction of the inequality symbol (>, <, ≥, ≤) when performing those operations.
- The solution to an inequality is a range of values that satisfy the inequality, rather than a single value or set of values.