Question

Find all values of \(x\) that satisfy both inequalities simultaneously. (a) \(3 x+7>1\) and \(2 x+1<3\) (b) \(3 x+7>1\) and \(2 x+1>-4\) (c) \(3 x+7>1\) and \(2 x+1<-4\)

Short answer

(a) \(-2 < x < 1\); (b) \(x > -2\); (c) No solution.

Step-by-step solution

Solve the first inequality for x in part (a)

We begin with the inequality given: \(3x + 7 > 1\).Subtract 7 from both sides to isolate terms involving \(x\):\(3x > 1 - 7\)\(3x > -6\).Next, divide both sides by 3:\(x > -2\).

Solve the second inequality for x in part (a)

Now, work with the second inequality:\(2x + 1 < 3\).Subtract 1 from both sides to isolate terms involving \(x\):\(2x < 3 - 1\)\(2x < 2\).Divide both sides by 2:\(x < 1\).

Combine solutions for part (a)

To solve part (a), both inequalities must be satisfied. Thus, we need \(x\) in the range:\(-2 < x < 1\).This represents all values that \(x\) can take.

Solve the first inequality for x in part (b)

For part (b), the first inequality is the same:\(3x + 7 > 1\),which simplifies to:\(x > -2\).

Solve the second inequality for x in part (b)

The second inequality becomes:\(2x + 1 > -4\).Subtract 1 from both sides:\(2x > -4 - 1\)\(2x > -5\).Divide both sides by 2:\(x > -\frac{5}{2}\).

Combine solutions for part (b)

Since both conditions must be met, take the intersection:\(x > -2\) and \(x > -\frac{5}{2}\).Thus, \(x > -2\) is already more restrictive, so the solution is: \(x > -2\).

Solve the first inequality for x in part (c)

For part (c), the first inequality remains the same:\(3x + 7 > 1\),which simplifies to:\(x > -2\).

Solve the second inequality for x in part (c)

Now, solve the second inequality:\(2x + 1 < -4\).Subtract 1 from both sides:\(2x < -4 - 1\)\(2x < -5\).Divide both sides by 2:\(x < -\frac{5}{2}\).

Combine solutions for part (c)

To satisfy both conditions:\(x > -2\) and \(x < -\frac{5}{2}\).However, since \(-\frac{5}{2} = -2.5\), there are no values of \(x\) that can satisfy both inequalities simultaneously.

Key concepts

Solving Inequalities

In mathematics, solving inequalities is a fundamental concept, much like solving equations. An inequality is simply a statement that compares two expressions using symbols like ">", "<", ">=", "<=", etc. The process of solving inequalities involves finding the set of values that make the inequality true. To solve an inequality, we perform operations similar to those used in solving equations. However, there's one key difference: if you multiply or divide both sides by a negative number, you must flip the inequality sign.

Here's a quick guide on how to solve an inequality:

• Identify the inequality and note the inequality symbol.
• Isolate the variable on one side using addition, subtraction, multiplication, or division.
• If you multiply or divide by a negative number, flip the inequality sign.
• Simplify the expression if necessary, and write down the solution.

Understanding solving inequalities is essential as it is widely applied in various mathematical problems and practical situations.

Compound Inequalities

Compound inequalities involve two or more inequalities joined by the words "and" or "or". The purpose is to find all values that satisfy all conditions simultaneously. These are common in math problems where two ranges of values are considered.

When solving compound inequalities, it's crucial to realize:

• A compound inequality using "and" means the solution must satisfy both conditions and is represented as the intersection of solutions.
• A compound inequality using "or" allows any value satisfying at least one condition and involves the union of solutions.

In the context of the original exercise, we deal primarily with "and" compound inequalities. For instance, in part (a), both conditions \(x > -2\) and \(x < 1\) combine to form the solution set \(-2 < x < 1\). This indicates the overlap, or intersection, of the ranges defined by the individual inequalities.

Inequality Solutions

Finding inequality solutions means determining the set or range of all possible values that make the inequality true. The solution is often expressed as an interval on a number line. Sometimes, if no value satisfies the inequality, we say that there is no solution.

The solution process involves the following steps:

• Break down each inequality within a compound inequality individually.
• Determine the solution for each part as a separate equation.
• Calculate the intersection or union, depending on the type of compound inequality.

For example, in exercise part (c), solving the inequalities \(x > -2\) and \(x < -\frac{5}{2}\) leads to an interesting result where there are no overlapping solutions, indicating no values of \(x\) can satisfy both conditions. This teaches us that not all compound inequalities will have a valid solution.