Solve the System of Inequalities X/4+4< 3 or -x+2< 2
The question presents two separate inequalities and asks you to find the range of values for the variable 'X' that satisfies at least one of them. The system of inequalities is not bound by the conjunction 'and', which would require both inequalities to be true simultaneously; instead, it uses the disjunction 'or', meaning a solution is valid if it satisfies either one of the inequalities or both. You are being asked to solve each inequality for 'X' and then take the union of the solution sets since 'or' implies that values in either solution set are acceptable.
$\frac{X}{4} + 4 < 3$or$- x + 2 < 2$
Answer
Solution:
Step:1
Refine the first inequality.
Step:1.1
Isolate the variable $X$ on one side of the inequality.
Step:1.1.1
Deduct $4$ from each side.$\frac{X}{4} < 3 - 4$ or $- x + 2 < 2$
Step:1.1.2
Compute $3 - 4$.$\frac{X}{4} < -1$ or $- x + 2 < 2$
Step:1.2
Amplify both sides by $4$.$\frac{X}{4} \times 4 < -1 \times 4$ or $- x + 2 < 2$
Step:1.3
Streamline the equation.
Step:1.3.1
Clarify the left-hand side.
Step:1.3.1.1
Eliminate the common factor of $4$.
Step:1.3.1.1.1
Remove the common factor.$X < -1 \times 4$ or $- x + 2 < 2$
Step:1.3.1.1.2
Restate the inequality.$X < -4$ or $- x + 2 < 2$
Step:1.3.2
Clarify the right-hand side.
Step:1.3.2.1
Multiply $-1$ by $4$.$X < -4$ or $- x + 2 < 2$
Step:2
Refine the second inequality.
Step:2.1
Isolate the variable $x$ on one side of the inequality.
Step:2.1.1
Deduct $2$ from each side.$X < -4$ or $- x < 2 - 2$
Step:2.1.2
Compute $2 - 2$.$X < -4$ or $- x < 0$
Step:2.2
Divide the inequality $- x < 0$ by $-1$ and reverse the inequality sign.
Step:2.2.1
Divide by $-1$.$X < -4$ or $\frac{- x}{- 1} > \frac{0}{- 1}$
Step:2.2.2
Simplify the left-hand side.
Step:2.2.2.1
Dividing negatives yields a positive.$X < -4$ or $x > 0$
Step:2.2.2.2
Simplify $x$ divided by $1$.$X < -4$ or $x > 0$
Step:2.2.3
Simplify the right-hand side.
Step:2.2.3.1
Divide $0$ by $-1$.$X < -4$ or $x > 0$
Step:3
The solution to the system of inequalities is $X < -4$ or $x > 0$.
Knowledge Notes:
When solving systems of inequalities, there are several key points to remember:
Isolating the Variable: It's often necessary to get the variable on one side of the inequality to clearly see the solution set.
Performing Operations: Just like equations, you can add, subtract, multiply, or divide both sides of an inequality by the same number, except when multiplying or dividing by a negative number, which reverses the inequality sign.
Combining Inequalities: When you have a system with "or," it means that solutions to either inequality are acceptable. This often results in a union of solution sets.
Graphical Representation: Inequalities can be represented on a number line, where a solution to an inequality is often shown as a ray or interval.
Checking Solutions: It's important to check that the solutions make sense by substituting them back into the original inequalities.
In LaTeX, to ensure proper rendering of mathematical expressions, use the appropriate symbols and commands. For example, fractions are written with the \frac{}{} command, inequalities use < and >, and multiplication can be indicated with \times or \cdot. Remember to enclose LaTeX expressions within dollar signs for inline rendering or double dollar signs for display mode.
