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.
Part A: To solve the equation ax−5=18, we need to isolate the variable x.
First, we add 5 to both sides of the equation to get:
ax = 18 + 5
ax = 23
Then, we divide both sides of the equation by a to solve for x:
x = 23/a
Therefore, the solution for the equation ax−5=18 is x = 23/a.
Part B: To solve the inequality ax−5>18, we follow similar steps as Part A but with the addition of considering the sign change when multiplying or dividing by a negative number.
First, we add 5 to both sides of the inequality to get:
ax > 18 + 5
ax > 23
Then, we divide both sides of the inequality by a, while considering the sign change:
If a > 0:
x > 23/a
If a < 0:
x < 23/a
Therefore, the solution for the inequality ax−5>18 will be either x > 23/a or x < 23/a, depending on the sign of a.
Part C: The difference between an equation and an inequality regarding solving steps and solutions lies in the nature of the mathematical statement.
An equation is a statement of equality, where the goal is to find the value(s) of the variable(s) that make both sides of the equation equal. The solving steps involve performing operations on both sides of the equation to isolate the variable.
An inequality, on the other hand, is a statement of inequality or comparison between two quantities. The goal is to find the range of values that satisfy the inequality. The solving steps involve similar operations as in solving equations, with the addition of considering the sign change when multiplying or dividing by a negative number. The solutions for an inequality can usually be expressed as a range of values that satisfy the inequality.
Part A: Solve the equation for x:
To solve the equation ax - 5 = 18, we will isolate x by performing the necessary 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 for the equation ax - 5 = 18 is x = 23/a.
Part B: Solve the inequality for x:
To solve the inequality ax - 5 > 18, we will follow these 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, be careful since dividing by a negative value would require flipping the inequality sign:
Dividing by a positive value: (No sign flip needed)
(ax)/a > 23/a
x > 23/a
Therefore, the solution for the inequality ax - 5 > 18 is x > 23/a.
Part C: The difference between an equation and an inequality regarding solving steps and solutions:
- Equations have equal signs (=), while inequalities have either greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) signs.
- When solving equations, the goal is to find the value of the variable that makes the equation true. There is usually only one solution.
- When solving inequalities, the goal is to find a range of values for the variable that makes the inequality true. There can be multiple solutions or a range of solutions.
- For equations, the steps involve performing operations to isolate the variable.
- For inequalities, the steps also involve performing operations, but the direction of the inequality sign might need to be flipped if dividing or multiplying by a negative number.