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)
0 / 10
Part A:
ax - 5 = 18
To solve for x, we need to isolate it on one side of the equation.
First, let's add 5 to both sides of the equation.
ax - 5 + 5 = 18 + 5
ax = 23
Next, we need to divide both sides of the equation by 'a' to solve for x.
ax/a = 23/a
x = 23/a
Therefore, the solution for x is x = 23/a.
Part B:
ax - 5 > 18
To solve the inequality, we need to isolate x on one side of the inequality.
First, let's add 5 to both sides of the inequality.
ax - 5 + 5 > 18 + 5
ax > 23
Next, we need to divide both sides of the inequality by 'a'. However, we need to consider the sign of 'a'.
If 'a' is positive, then we can divide both sides by 'a' without changing the direction of the inequality.
If 'a' is negative, then we need to flip the direction of the inequality when dividing by 'a'.
Since we don't know the sign of 'a' in this problem, we'll leave the inequality as ax > 23.
Therefore, the solution for x is x > 23/a, with the direction of the inequality depending on the sign of 'a'.
Part C:
The difference between an equation and an inequality regarding solving steps and solutions is that an equation seeks to find a specific value or values for the variable(s) to make the equation true, whereas an inequality seeks to find a range or set of values for the variable(s) to make the inequality true.
In solving an equation, we aim to isolate the variable and determine the exact value(s) that satisfy the equation.
In solving an inequality, we aim to isolate the variable and determine the range or set of values that satisfy the inequality. The solutions for an inequality often involve a range of possible values instead of a single value. Additionally, when dividing or multiplying both sides of an inequality by a negative number, the direction of the inequality must be flipped.
Part A: Solve the equation ax - 5 = 18
To solve this equation, we need to isolate the variable x.
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 to solve for x.
(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 ax - 5 > 18
To solve this inequality, we will also isolate the variable x.
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, but since we don't know if a is positive or negative, we need to consider two scenarios.
Scenario 1: a > 0
If a is positive, we can divide both sides by a without changing the direction of the inequality.
ax/a > 23/a
x > 23/a
Scenario 2: a < 0
If a is negative, we must divide both sides by a and reverse the direction of the inequality.
ax/a < 23/a
x < 23/a
Therefore, if a is positive, the solution for the inequality ax - 5 > 18 is x > 23/a, and if a is negative, the solution is x < 23/a.
Part C: The difference between an equation and an inequality regarding solving steps and solutions is that an equation aims to find a specific value(s) for the variable, while an inequality finds a range of values for the variable that satisfy the given conditions.
In solving an equation, we perform various operations (addition, subtraction, multiplication, division) to isolate the variable and determine its value(s). We aim to find a specific solution or set of solutions that make the equation true.
In solving an inequality, we also perform similar operations to isolate the variable, but the solution results in a range of values that make the inequality true. The solution can be represented as an interval or as a combination of inequalities (greater than, less than, greater than or equal to, less than or equal to).