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 for x, we need to isolate x on one side of the equation.
Starting with the given equation:
ax−5=18
We want to get rid of the -5 on the left side of the equation, so we add 5 to both sides:
ax−5+5=18+5
ax=23
To solve for x, we divide both sides of the equation by a:
(ax)/a = 23/a
x = 23/a
So the solution to the equation ax−5=18 is x = 23/a.
Part B:
To solve the inequality ax−5>18 for x, we need to isolate x on one side of the inequality, but this time we also need to pay attention to the direction of the inequality sign.
Starting with the given inequality:
ax−5>18
First, we add 5 to both sides of the inequality:
ax−5+5>18+5
ax>23
Now, we divide both sides of the inequality by a. However, we need to consider two cases separately depending on the value of a:
1) If a > 0, we can divide by a without changing the direction of the inequality:
(ax)/a > 23/a
x > 23/a
2) If a < 0, we need to flip the direction of the inequality when dividing by a:
(ax)/a < 23/a
x < 23/a
So, the solution to the inequality ax−5>18, depending on the value of a, is:
- If a > 0, x > 23/a
- If a < 0, x < 23/a
Part C:
The difference between an equation and an inequality lies in the number of possible solutions.
An equation has a single solution or a finite number of solutions that make the equation true. When solving an equation, the steps involve manipulating the equation to isolate the variable and find the specific value(s) that satisfy the equation.
On the other hand, an inequality has an infinite number of solutions since it represents a range of values. When solving an inequality, the steps involve determining the range of values for the variable that make the inequality true. The solution to an inequality is usually expressed in terms of intervals or inequalities.
TY :DD
Part A: Solve the equation for x (ax - 5 = 18):
Step 1: Start with the equation: ax - 5 = 18.
Step 2: Add 5 to both sides of the equation to isolate the variable term: ax = 18 + 5.
Step 3: Simplify: ax = 23.
Step 4: 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: Solve the inequality for x (ax - 5 > 18):
Step 1: Start with the inequality: ax - 5 > 18.
Step 2: Add 5 to both sides of the inequality to isolate the variable term: ax - 5 + 5 > 18 + 5.
Step 3: Simplify: ax > 23.
Step 4: Divide both sides of the inequality by 'a', but consider the sign change if 'a' is negative.
If 'a' is positive, then x > 23/a.
If 'a' is negative, then x < 23/a.
Therefore, the solution for the inequality ax - 5 > 18 depends on the value of 'a' and can be expressed as x > 23/a or x < 23/a.
Part C: The difference between an equation and an inequality regarding solving steps and solutions lies in the nature of their solutions.
In an equation, we seek to find a value or values of the variable that make the equation true. This value is typically represented by a single solution or a set of solutions. The solving steps involve manipulating the equation to isolate the variable and determine its value(s).
In an inequality, however, we seek to find a range of values that satisfy the given inequality. The solution may be represented by an interval or multiple intervals, depending on the nature of the inequality. The solving steps involve manipulating the inequality to isolate the variable and determine the valid range(s) of values.
Thus, while both equations and inequalities involve solving for the variable, the difference lies in the nature of their solutions and the resulting steps required to find those solutions.
Part A: Solve the equation ax - 5 = 18 showing all steps for credit.
To solve the equation ax - 5 = 18, we want to isolate the variable x on one side of the equation. Here's how we can do that:
Step 1: Add 5 to both sides of the equation to get rid of the constant term -5:
ax - 5 + 5 = 18 + 5
This simplifies to: ax = 23
Step 2: Divide both sides of the equation by the coefficient a to solve for x:
(ax) / a = 23 / a
The coefficient a cancels out on the left side, leaving us with: x = 23 / a
So the solution to the equation ax - 5 = 18 is x = 23 / a.
Part B: Solve the inequality ax - 5 > 18 showing all steps for credit.
To solve the inequality ax - 5 > 18, we follow similar steps as solving an equation but with one additional consideration when dealing with inequalities.
Step 1: Add 5 to both sides of the inequality:
ax - 5 + 5 > 18 + 5
Simplifying gives us: ax > 23
Step 2: Divide both sides of the inequality by the coefficient a, but with a slight variation depending on the sign of a.
- If a is positive, we divide by a without changing the direction of the inequality:
(ax) / a > 23 / a
Resulting in: x > 23 / a
- If a is negative, dividing by a requires you to flip the direction of the inequality:
(ax) / a < 23 / a
Resulting in: x < 23 / a
So the solution to the inequality ax - 5 > 18, depending on the sign of a, is either x > 23 / a or x < 23 / a.
Part C: State the difference between an equation and an inequality regarding solving steps and solutions.
An equation is a statement of equality between two expressions, usually involving an = sign, representing that the two sides of the equation are balanced or have the same value. When solving an equation, our goal is to find the numerical value of the variable(s) that makes the equation true.
An inequality, on the other hand, is a statement of inequality between two expressions, usually involving symbols such as > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to). When solving an inequality, we need to determine the range of values for the variables that make the inequality true.
In terms of solving steps, equations and inequalities follow similar procedures, like isolating the variable. However, when solving inequalities, we have to be careful about the direction of the inequality sign and make appropriate adjustments when dividing or multiplying by a negative number.
Regarding solutions, equations typically have a single solution, or sometimes multiple solutions, that make the equation true. Inequalities, on the other hand, often have a range of solutions, represented by an interval or combination of intervals that satisfy the given inequality. This is due to the fact that an inequality represents a range of possible values rather than a single value.