Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the numbers \(x\) that satisfy both of the inequalities $$ 3 x+4 \leq 7 \text { and } 5-2 x<13 $$

Short answer

-4 < x \leq 1

Step-by-step solution

- Solve the first inequality

Start with the first inequality: \[3x + 4 \leq 7\]Subtract 4 from both sides to isolate the term with x: \[3x \leq 3\]Divide by 3 to solve for x: \[x \leq 1\]

- Solve the second inequality

Now, solve the second inequality: \[5 - 2x < 13\]Subtract 5 from both sides to isolate the term with x: \[-2x < 8\]Divide both sides by -2, and remember to reverse the inequality sign when dividing by a negative number: \[x > -4\]

- Combine the solutions

Combine the solutions from both inequalities. The first inequality gives: \[x \leq 1\]The second inequality gives: \[x > -4\]Therefore, the combined solution is: \[-4 < x \leq 1\]

Key concepts

Linear Inequalities

Linear inequalities involve expressions that use inequality signs like \(<\), \(\leq\), \(>\), or \(\geq\) instead of an equal sign. The goal is to determine the range of values for the variable that makes the inequality true.
A typical linear inequality can be written in the form \(ax + b < c\), where we solve for the variable \(x\) just like in linear equations, but we must be careful with the inequality direction.
When solving, remember these key steps:

• Perform the same operations on both sides of the inequality.
• If you multiply or divide by a negative number, reverse the inequality sign.
• Graphically, an inequality doesn't have a single solution but a range of values.

Understanding these principles is critical in solving linear inequalities efficiently.

Inequality Solution

Finding the solution to an inequality involves isolating the variable on one side of the inequality. For example, in the inequality \((3x + 4 \leq 7)\), we want to isolate \(x\).
1. Subtract 4 from both sides:
\(3x + 4 - 4 \leq 7 - 4) \rightarrow 3x \leq 3\)
2. Divide by 3:
\(\frac{3x}{3} \leq \frac{3}{3}) \rightarrow x \leq 1\)
This tells us that any number less than or equal to 1 satisfies the inequality.
The same logic applies to more complex inequalities, stripping away constants and coefficients until you isolate the variable. When dealing with negative multipliers, always note the rule about reversing the inequality sign.
Using these steps will consistently help you find the range of values that satisfy your inequality.

Step-by-Step Problem Solving

Approaching inequalities methodically helps in avoiding errors. Let's solve the given example step-by-step:
**Step 1:** Start with \(3x + 4 \leq 7\)
Subtract 4 from both sides:
\(3x \leq 3\).
Divide by 3:
\(x \leq 1\)
We find that \(x\) can be any number up to and including 1.
**Step 2:** Now, consider \(5 - 2x < 13\)
Subtract 5 from both sides:
\(-2x < 8\).
Divide by -2 and reverse the sign:
\(x > -4\)
This inequality shows \(x\) must be greater than -4.
**Step 3:** Combine the two solutions:
From \(x \leq 1\) and \(x > -4\), the combined solution is:
\(-4 < x \leq 1\)
Practicing these steps ensures that solving inequalities becomes a repeatable and accurate process.

Combining Inequalities

Combining inequalities means finding the common values that satisfy all given inequalities. Suppose we have: \(3x + 4 \leq 7\) and \(5 - 2x < 13\). Solve the inequalities separately first:
**Inequality 1:**
\(3x + 4 \leq 7\)
Subtract 4:
\(3x \leq 3\).
Divide by 3:
\(x \leq 1\)
**Inequality 2:**
\(5 - 2x < 13\)
Subtract 5:
\(-2x < 8\).
Divide by -2 and flip the sign:
\(x > -4\)
We need the combined range where both conditions hold: \(x \leq 1\) AND \(x > -4\).
Therefore, the combined solution is:
\(-4 < x \leq 1\)
By plotting on a number line or using logical intersection of the ranges, visualize the common solution. Always check edge cases to ensure the logic holds.