Let $f$ and $g$ be real functions defined by $f(x) = 2x + 1$ and $g(x) = 4x - 7$. (a) For what real numbers $x$, $f(x) = g(x)$? (b) For what real numbers $x$, $f(x) < g(x)$?

> [!TLDR] > We find the specific $x$ where the functions $f(x)$ and $g(x)$ are equal by solving a linear equation, and the range of $x$ where $f(x)$ is less than $g(x)$ by solving a linear inequality. The functions are equal at $x=4$, and $f(x) 4$. When we are given two functions, $f(x)$ and $g(x)$, and asked to compare them, we are essentially looking for the input values $x$ that satisfy certain conditions regarding their outputs. * **$f(x) = g(x)$** means we are looking for the specific $x$-value(s) where the output of $f$ is exactly the same as the output of $g$. Graphically, this corresponds to finding the $x$-coordinate(s) of the intersection point(s) of their graphs. * **$f(x) < g(x)$** means we are looking for the range of $x$-values where the output of $f$ is strictly less than the output of $g$. Graphically, this corresponds to finding the $x$-values where the graph of $f(x)$ lies *below* the graph of $g(x)$. Since $f(x)$ and $g(x)$ are linear functions, their graphs are straight lines. Two distinct lines can intersect at most once. This means there will be at most one $x$-value where $f(x) = g(x)$. This intersection point then divides the number line into regions where one function is greater or less than the other. Let's solve each part. ### Part (a): For what real numbers $x$, $f(x) = g(x)$? 1. **Set up the equation:** We are given $f(x) = 2x + 1$ and $g(x) = 4x - 7$. To find when $f(x) = g(x)$, we set their expressions equal to each other: $$2x + 1 = 4x - 7$$ 2. **Solve for $x$:** This is a standard linear equation. Our goal is to isolate $x$ on one side of the equation. * Subtract $2x$ from both sides to gather $x$ terms: $$1 = (4x - 2x) - 7$$ $$1 = 2x - 7$$ * Add $7$ to both sides to gather constant terms: $$1 + 7 = 2x$$ $$8 = 2x$$ * Divide both sides by $2$: $$\frac{8}{2} = x$$ $$4 = x$$ 3. **Conclusion:** The functions $f(x)$ and $g(x)$ are equ

Let f and g be real functions defined by f(x) = 2x – Class 11 Mathematics

NCERT Exemplar · Q12

f

(a) For what real numbers

x

f(x) = g(x)

(b) For what real numbers

x

f(x) < g(x)

Uttarakhand UbseShort· 3mImportance★★★★★

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Concept understanding — Linear Inequality Solutions

Linear Inequality Solutions – A First Look

Imagine you're standing on a number line. You know exactly where the number 5 is. But what if I asked you to stand on "all numbers greater than 5"? You can't stand on all of them at once — they stretch infinitely to the right. That's the core idea of an inequality: instead of one exact point, you get a whole region of possible values.

A linear inequality is just like a linear equation ( $ax + b = 0$ ), but instead of an equals sign, you have one of these: $<$ , $>$ , $\le$ , or $\ge$ . The solution is not a single number — it's an interval (or a union of intervals) on the number line.

From Equation to Inequality

Start with a simple equation:

2x + 3 = 7

Solve it: $2x = 4 \implies x = 2$ . One point.

Now change it to an inequality:

2x + 3 > 7

Solve it the same way — but the meaning changes. Subtract 3: $2x > 4$ . Divide by 2: $x > 2$ .

The solution is all numbers greater than 2. On a number line, you draw an open circle at 2 (because 2 itself is not included) and shade everything to the right.

Watch out

If you multiply or divide both sides of an inequality by a negative number, the inequality sign reverses. For example: $-x < 5$ becomes $x > -5$ . This is the single most common mistake students make.

The Four Types of Solutions

Inequality	Meaning	Number line representation
$x > a$	All numbers strictly greater than $a$	Open circle at $a$ , shade right
$x \ge a$	All numbers greater than or equal to $a$	Closed (filled) circle at $a$ , shade right
$x < a$	All numbers strictly less than $a$	Open circle at $a$ , shade left
$x \le a$	All numbers less than or equal to $a$	Closed circle at $a$ , shade left

The solution set is usually written in interval notation:

$x > 2$ → $(2, \infty)$
$x \le -3$ → $(-\infty, -3]$

Parentheses $($ or $)$ mean the endpoint is not included. Brackets $[$ or $]$ mean it is included.

Solving a Linear Inequality: Step by Step

Solve $3x - 5 \le 7x + 3$ .

Bring variable terms to one side:

$3x - 5 - 7x \le 3$

$-4x - 5 \le 3$
Isolate the variable term:

$-4x \le 8$
Divide by the coefficient (here it's $-4$ , so reverse the sign):

$x \ge -2$

The solution is $x \ge -2$ , or in interval notation: $[-2, \infty)$ .

Tip

Always check your answer by testing a number from the solution set. For $x \ge -2$ , test $x = 0$ : $3(0)-5 = -5$ and $7(0)+3 = 3$ . Is $-5 \le 3$ ? Yes. Now test a number outside, say $x = -3$ : $3(-3)-5 = -14$ and $7(-3)+3 = -18$ . Is $-14 \le -18$ ? No — so the inequality fails, confirming our solution is correct.

Why This Matters …

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We find the specific $x$ where the functions $f(x)$ and $g(x)$ are equal by solving a linear equation, and the range of $x$ where $f(x)$ is less than $g(x)$ by solving a linear inequality. The functions are equal at $x=4$ , and $f(x) < g(x)$ for all $x > 4$ .

When we are given two functions, $f(x)$ and $g(x)$ , and asked to compare them, we are essentially looking for the input values $x$ that satisfy certain conditions regarding their outputs.

$f(x) = g(x)$ means we are looking for the specific $x$ -value(s) where the output of $f$ is exactly the same as the output of $g$ . Graphically, this corresponds to finding the $x$ -coordinate(s) of the intersection point(s) of their graphs.
$f(x) < g(x)$ means we are looking for the range of $x$ -values where the output of $f$ is strictly less than the output of $g$ . Graphically, this corresponds to finding the $x$ -values where the graph of $f(x)$ lies below the graph of $g(x)$ .

Since $f(x)$ and $g(x)$ are linear functions, their graphs are straight lines. Two distinct lines can intersect at most once. This means there will be at most one $x$ -value where $f(x) = g(x)$ . This intersection point then divides the number line into regions where one function is greater or less than the other.

Let's solve each part.

Part (a): For what real numbers $x$ , $f(x) = g(x)$ ?

Set up the equation: We are given $f(x) = 2x + 1$ and $g(x) = 4x - 7$ . To find when $f(x) = g(x)$ , we set their expressions equal to each other:

2x + 1 = 4x - 7

Solve for $x$ : This is a standard linear equation. Our goal is to isolate $x$ $x$ on one side of the equation.
- Subtract $2x$ from both sides to gather $x$ terms:

1 = (4x - 2x) - 7

1 = 2x - 7

*   Add $7$ to both sides to gather constant terms:

1 + 7 = 2x

8 = 2x

*   Divide both sides by $2$:

\frac{8}{2} = x

4 = x

Conclusion: The functions $f(x)$ and $g(x)$ are equal when $x = 4$ .

Part (b): For what real numbers $x$ , $f(x) < g(x)$ ?

Set up the inequality: We want to find when $f(x)$ is strictly less than $g(x)$ . We use the given expressions to form an inequality: $2x + 1 < 4x - 7$ …

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Q.Let fff and ggg be real functions defined by f(x)=2x+1f(x) = 2x + 1f(x)=2x+1 and g(x)=4x−7g(x) = 4x - 7g(x)=4x−7.

Part (a): For what real numbers xxx, f(x)=g(x)f(x) = g(x)f(x)=g(x)?

Part (b): For what real numbers xxx, f(x)<g(x)f(x) < g(x)f(x)<g(x)?

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Q.Let $f$ and $g$ be real functions defined by $f(x) = 2x + 1$ and $g(x) = 4x - 7$ .

Part (a): For what real numbers $x$ , $f(x) = g(x)$ ?

Part (b): For what real numbers $x$ , $f(x) < g(x)$ ?

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